Preface to the First Edition
This book could be titled Applied Bond Math or, perhaps, Practical Bond Math. Those who do serious research on fixed-income securities and markets know that this subject matter goes far beyond the mathematics covered herein. Those who are interested in discussions about “pricing kernels” and “stochastic discount rates” will have to look elsewhere. My target audience is those who work in the finance industry (or aspire to), know what a Bloomberg page is, and in the course of the day might hear or use terms such as “yield to maturity,” “forward curve,” and “modified duration.”
My objective in Bond Math is to explain the theory and assumptions that lie behind the commonly used statistics regarding the risk and return on bonds. I show many of the formulas that are used to calculate yield and duration statistics and, in the Technical Appendix, their formal derivations. But I do not expect a reader to actually use the formulas or do the calculations. There is much to be gained by recognizing that “there exists an equation” and becoming more comfortable using a number that is taken from a Bloomberg page, knowing that the result could have been obtained using a bond math formula.
This book is based on my 25 years of experience teaching this material to graduate students and finance professionals. For that, I thank the many deans, department chairs, and program directors at the Boston University School of Management who have allowed me to continue teaching fixed- income courses over the years. I thank Euromoney Training in New York and Hong Kong for organizing four-day intensive courses for me all over the world. I thank training coordinators at Chase Manhattan Bank (and its heritage banks, Manufacturers Hanover and Chemical), Lehman Brothers, and the Bank of Boston for paying me handsomely to teach their employees on so many occasions in so many interesting venues. Bond math has been very, very good to me.
The title of this book emanates from an eponymous two-day course I taught many years ago at the old Manny Hanny. (Okay, I admit that I have always wanted to use the word “eponymous”; now I can cross that off my bucket list.) I thank Keith Brown of the University of Texas at Austin, who co-designed and co-taught many of those executive training courses, for emphasizing the value of relating the formulas to results reported on Bloomberg. I have found that users of “black box” technologies find comfort in knowing how those bond numbers are calculated, which ones are useful, which ones are essentially meaningless, and which ones are just wrong.
Our journey through applied and practical bond math starts in the money market, where we have to deal with anachronisms like discount rates and a 360-day year. A key point in Chapter 1 is that knowing the periodicity of an annual interest rate (i.e., the assumed number of periods in the year) is critical. Converting from one periodicity to another – for instance, from quarterly to semiannual – is a core bond math calculation that I use throughout the book. Money market rates can be deceiving because they are not intuitive and do not follow classic time-value-of-money principles taught in introductory finance courses. You have to know what you are doing to play with T-bills, commercial paper, and bankers acceptances.
Chapters 2 and 3 go deep into calculating prices and yields, first on zero-coupon bonds to get the ideas out for a simple security like U.S. Treasury STRIPS (i.e., just two cash flows) and then on coupon bonds for which coupon reinvestment is an issue. The yield to maturity on a bond is a summary statistic about its cash flows – it's important to know the assumptions that underlie this widely quoted measure of an investor's rate of return and what to do when those assumptions are untenable. I decipher Bloomberg's Yield Analysis page for a typical corporate bond, showing the math behind “street convention,” “U.S. government equivalent,” and “true” yields. The problem is distinguishing between yields that are pure data (and can be overlooked) and those that provide information useful in making a decision about the bond.
Chapter 4 continues the exploration of rate-of-return measures on an after-tax basis for corporate, Treasury, and municipal bonds. Like all tax matters, this necessarily gets technical and complicated. Taxation, at least in the U.S., depends on when the bond was issued (there were significant changes in the 1980s and 1990s), at what issuance price (there are different rules for original issue discount bonds), and whether a bond issued at (or close to) par value is later purchased at a premium or discount. Given the inevitability of taxes, this is important stuff – and it is stuff on which Bloomberg sometimes reports a misleading result, at least for U.S. investors.
Yield curve analysis, in Chapter 5, is arguably the most important topic in the book. There are many practical applications arising from bootstrapped implied zero-coupon (or spot) rates and implied forward rates – identifying arbitrage opportunities, obtaining discount factors to get present values, calculating spreads, and pricing and valuing derivatives. However, the operative assumption in this analysis is “no arbitrage” – that is, transactions costs and counterparty credit risk are sufficiently small so that trading eliminates any arbitrage opportunity. Therefore, while mathematically elegant, yield curve analysis is best applied to Treasury securities and LIBOR-based interest rate derivatives for which the no-arbitrage assumption is reasonable.
Duration and convexity, the subject of Chapter 6, is the most mathematical topic in this book. These statistics, which in classic form measure the sensitivity of the bond price to a change in its yield to maturity, can be derived with algebra and calculus. Those details are relegated to the Technical Appendix. Another version of the risk statistics measures the sensitivity of the bond price to a shift in the entire Treasury yield curve. I call the former yield and the latter curve duration and convexity and demonstrate where and how they are presented on Bloomberg pages.
Chapters 7 and 8 examine floating-rate notes (floaters), inflation-indexed bonds (linkers), and interest rate swaps. The idea is to use the bond math toolkit – periodicity conversions, bond valuation, after-tax rates of return, implied spot rates, implied forward rates, and duration and convexity – to examine securities other than traditional fixed-rate and zero-coupon bonds. In particular, I look for circumstances of negative duration, meaning market value and interest rates are positively correlated. That's an obvious feature for one type of interest rate swap but a real oddity for a floater and a linker.
Understanding the risk and return characteristics for an individual bond is easy compared to a portfolio of bonds. In Chapter 9, I show different ways of getting summary statistics. One is to treat the portfolio as a big bundle of cash flow and derive its yield, duration, and convexity is if it were just a single bond with many variable payments. While that is theoretically correct, in practice portfolio statistics are calculated as weighted averages of those for the constituent bonds. Some statistics can be aggregated in this manner and provide reasonable estimates of the “true” values, depending on how the weights are calculated and on the shape of the yield curve.
Chapter 10 is on bond strategies. If your hope is that I'll show you how to get rich by trading bonds, you'll be disappointed. My focus is on how the bond math tools and the various risk and return statistics that we can calculate for individual bonds and portfolios can facilitate either aggressive or passive investment strategies. I'll discuss derivative overlays, immunization, and liability-driven investing and conclude with a request that the finance industry create target-duration bond funds.
I'd like to thank my Wiley editors for allowing me to deviate from their usual publishing standards so that I can use in this book acronyms, italics, and notation as I prefer. Now let's get started in the money market.