Bond Math

The Theory Behind the Formulas

Donald J. Smith

• Publisher: Bloomberg Press
• Year: 2014
• ISBN:
  9781118866368
• Edition: 2
• Topic: Business
• Subtopic: Investments & Securities
• Index: Business

A bond calculation quick reference, complete with context and application insights

Bond Math is a quick and easy resource that puts the intricacies of bond calculations into a clear and logical order. This simple, readable guide provides a handy reference, teaching the reader how to think about the essentials of bond math. Much more than just a book of formulas, the emphasis is on how to think about bonds and the associated math, with plenty of examples, anecdotes, and thought-provoking insights that sometimes run counter to conventional wisdom. This updated second edition includes popular Bloomberg pages used in fixed-income analysis, including the Yield and Spread Analysis page, plus a companion website complete with an Online Workbook of multiple choice questions and answers and spreadsheet exercises. Detailed coverage of key calculations, including thorough explanations, provide practical guidance to working bond professionals.

The bond market is the largest and most liquid in the world, encompassing everything from Treasuries and investment grade corporate paper to municipals and junk bonds, trading over $900 billion daily in the U.S. alone. Bond Math is a guide to the inevitable calculations involved in managing bonds, with expert insight on the portfolios and investment strategies that puts the math in perspective. Clear and concise without sacrificing detail, this book helps readers to:

• Delineate the characteristics of different types of debt securities
• Calculate implied forward and spot rates and discount factors
• Work with rates of return, yield statistics, and interest rate swaps
• Understand duration-based risk measures, and more

Memorizing formulas is one thing, but really learning how to mentally approach the math behind bonds is something else entirely. This approach places calculations in context, and enables easier transition from theory to application. For the bond professional seeking a quick math reference, Bond Math provides that and so much more.

CHAPTER 1
Money Market Interest Rates

An interest rate is a summary statistic about the cash flows on a debt security such as a loan or a bond. As a statistic, it is a number that we calculate. An objective of this chapter is to demonstrate that there are many ways to do this calculation. Like many statistics, an interest rate can be deceiving and misleading. Nevertheless, we need interest rates to make financial decisions about borrowing and lending money and about buying and selling securities. To avoid being deceived or misled, we need to understand how interest rates are calculated.

It is useful to divide the world of debt securities into short-term money markets and long-term bond markets. The former is the home of money market instruments such as Treasury bills, commercial paper, bankers acceptances, bank certificates of deposit, and overnight and term sale-repurchase agreements (called “repos”). The latter is where we find coupon-bearing notes and bonds that are issued by the Treasury, corporations, federal agencies, and municipalities. The key reference interest rate in the U.S. money market is 3-month LIBOR (the London Interbank offer rate); the benchmark bond yield is on 10-year Treasuries.

This chapter is on money market interest rates. Although the money market usually is defined as securities maturing in one year or less, much of the activity is in short-term instruments, from overnight out to six months. The typical motivation for both issuers and investors is cash management arising from the mismatch in the timing of revenues and expenses. Therefore, primary investor concerns are liquidity and safety. The instruments themselves are straightforward and entail just two cash flows, the purchase price and a known redemption amount at maturity.

Let’s start with a practical money market investment problem. A fund manager has about $1 million to invest and needs to choose between two 6-month securities: (1) commercial paper (CP) quoted at 3.80% and (2) a bank certificate of deposit (CD) quoted at 3.90%. Assuming that the credit risks are the same and any differences in liquidity and taxation are immaterial, which investment offers the better rate of return, the CP at 3.80% or the CD at 3.90%? To the uninitiated, this must seem like a trick question—surely, 3.90% is higher than 3.80%. If we are correct in our assessment that the risks are the same, the CD appears to pick up an extra 10 basis points. The initiated know that first it is time for a bit of bond math.

INTEREST RATES IN TEXTBOOK THEORY

You probably were first introduced to the time value of money in college or in a job training program using equations such as these:

(1.1)

where FV = future value, PV = present value, i = interest rate per time period, and N = number of time periods to maturity.

The two equations are the same and merely are rearranged algebraically. The future value is the present value moved forward along a time trajectory representing compound interest over the N periods; the present value is the future value discounted back to day zero at rate i per period.

In your studies, you no doubt worked through many time-value-of-money problems, such as: How much will you accumulate after 20 years if you invest $1,000 today at an annual interest rate of 5%? How much do you need to invest today to accumulate $10,000 in 30 years assuming a rate of 6%? You likely used the time-value-of-money keys on a financial calculator, but you just as easily could have plugged the numbers into the equations in 1.1 and solved via the arithmetic functions.

The interest rate in standard textbook theory is well defined. It is the growth rate of money over time—it describes the trajectory that allows $1,000 to grow to $2,653 over 20 years. You can interpret an interest rate as an exchange rate across time. Usually we think of an exchange rate as a trade between two currencies (e.g., a spot or a forward foreign exchange rate between the U.S. dollar and the euro). An interest rate tells you the amount...