A whole is worth the sum of its parts. Even the most complex structured bond, credit arbitrage strategy or hedge trade can be broken down into its component parts, and if we understand the elemental components, we can then value the whole as the sum of its parts. We can quantify the risk that is hedged and the risk that is left as the residual exposure. If we learn to view all financial trades and securities as engineered packages of building blocks, then we can analyze in which structures some parts may be cheap and some may be rich. It is this relative value arbitrage principle that drives all modern trading and investment.

This book is an easy-to-understand guide to the complex world of today’s financial markets teaching you what money and capital markets are about through a sequence of arbitrage-based numerical illustrations and exercises enriched with institutional detail. Filled with insights and real life examples from the trading floor, it is essential reading for anyone starting out in trading.

Using a unique structural approach to teaching the mechanics of financial markets, the book dissects markets into their common building blocks: spot (cash), forward/futures, and contingent (options) transactions. After explaining how each of these is valued and settled, it exploits the structural uniformity across all markets to introduce the difficult subjects of financially engineered products and complex derivatives.

The book avoids stochastic calculus in favour of numeric cash flow calculations, present value tables, and diagrams, explaining options, swaps and credit derivatives without any use of differential equations.

**ROBERT DUBIL** has been an Associate Professor in the finance department at the University of Utah since 2005. Prior to this he was Chief Strategist at HedgeStreet where he also wrote a blog as Dr Bob, and has held positions at UBS as Head of Quantitative Research and Fixed Income Options Trading; Chase Manhattan as Head of Exotics; Merrill Lynch as a Fixed Income Derivatives Trader, and latter as Director of Analytics in the Corporate Risk Management Group; Nomura; and J.P. Morgan. Professor Dubil holds a PhD and MBA from the University of Connecticut and an MA from Wharton. He published *An Arbitrage Guide to Financial Markets* (John Wiley & Sons, Ltd) in 2004 and has written a number of book chapters and articles on liquidity, derivatives and personal finance that have appeared in the *Journal of Applied Finance*, *Financial Services Review*, *Journal of Wealth Management*, *Journal of Investing*, and the *Journal of Financial Planning*. In Robert’s spare time he enjoys piano, skiing the greatest snow on earth and tennis. His second serve could use a lot of improvement though.

2

Spot Markets

The art of present valuing, otherwise known as Bond Math, is the basis of all security valuation, and the majority of this chapter will be devoted to Bond Math. We abstract from the reality of the many types of borrowers in the world such as sovereign governments, corporations, and municipalities, each of which is subject to different regulations and taxes, and each has its own credit worthiness – all of which affect yield levels in each market. We assume that we are within one of these markets with one issuer and a set of bonds issued by that issuer. We introduce the most basic bond structures – zero-coupon, coupon, amortizing, and floating rate – which are going to serve as building blocks for all other concepts. We go through simple annual calculations, then introduce real-life complications: compounding more frequently than annually, day counting for interest calculations and accruals. We then arrive at the fundamental concept of the discount curve or the term structure of discount or zero-coupon rates. Using the arbitrage principle that the whole must equal the sum of its parts (in value) for all the bonds in any given market, we build one set of discount rates or factors that allow us to value any new bond in that market. The value of a bond with arbitrary cash flows is equal to the sum of all its cash flows multiplied by the correct spot discount factors. This is not because our math says so, but because the greed of financial traders will not allow the same item, repackaged under a disguised name, to trade at a different value. Any bond can be shown to be a package of positions in bonds already existing in the market, and the bond's value can be computed from the sum of these positions. In general, some of these positions can be long (bought) and some short. At the end of the chapter, when we cover equity, commodity, and currency markets, we highlight the important concept of shorting securities. With that, our overview of the spot transactions as the most basic building blocks in financial engineering is complete.

2.1 BONDS AND ANNUAL BOND MATH

In this book, we use the term “loan” and “bond” interchangeably. A loan is a private credit arrangement between a borrower and a lender in which the lender acquires rights to future cash flows from the borrower. A lender, subsequently, can privately resell the loan to another willing lender. A bond is a publicly traded credit arrangement. The issuer borrows money by selling the bond to a distributed group of creditors who can then resell their portion of the issued bonds to other willing creditors.

Bonds that do not contain options are easy to analyze and value. They typically come in one of four forms: zero-coupon (discount) bonds, coupon bonds, amortizing bonds, and floating rate bonds.

2.1.1 Zero-Coupon Bond

Jack lends money to the ABC Company by buying a 4-year $1,000 face value zero-coupon bond. Let us suppose the interest rate (called the yield-to-maturity) Jack is going to earn is 5%. Figure 2.1 illustrates how we can value the bond by discounting the cash flow back through time.

Figure 2.1 A 4-year zero-coupon bond

In Figure 2.1 Jack is lending $822.70 to ABC today (the price he pays) and acquires a claim to receive $1,000 in 4 years’ time. Jack can resell this claim later for an amount that will depend on the interest rate at the time. The buyer will perform a backward discounting calculation similar to that shown in the figure, but only using the remaining number of years to maturity. Only if the interest rate does not change and stays at 5% throughout the life of the loan will the amounts shown in the dashed boxes represent the actual future prices of the bond.

Zero-coupon bonds are typically issued with face values (par values) that are round numbers like $1,000, $5,000, or $100,000. They are bought at a discount from par and the interest is imputed in the difference between the price paid (usually not a round number) and the face value received at maturity or the price received at the time of the resale. The price is typically quoted as the percent of par. In our example, the price would be quoted as 82.27. Note that the percent price converted to a fraction 0.8227 represents the value today of $1 received 4 years from now. The number 0.8227 is called the discount factor.

In general, bond and loan repayment values do not have to be round numbers. Suppose Jack was willing to lend $1,035.46 to ABC. How much should he expect 4 years from now if he earns 5%? Figure 2.2 illustrates the answer. This time we sweep forward starting with the known amount of the loan, i.e., the purchase price of the bond.

Figure 2.2 A $1,035.46 zero-coupon loan earning 5%

In general, the price of the zero-coupon bond, expressed as a percentage of par, is equal to:

where r is the interest rate and n is the number of years. The spot discount factor (df) is defined as:

The discount factor will prove to be extremely important when we come to value other securities. Each security will be treated as a claim on a stream of cash flows at different times. We will find the present value of those cash flows by multiplying them by their respective discount factors and summing the products to get the total value of the security. The zero-coupon bond is the only bond that has only one cash flow. All other bonds have multiple cash flows.

2.1.2 Coupon Bond

A coupon bond pays a periodic stream of identical cash flows (called coupon interest) and a one-time cash flow at maturity (called the par or face value). Suppose Jack buys a 4-year 6% annual coupon $1,000 par value bond issued by the ABC Company to earn a 5% annual interest rate (yield-to-maturity) on his investment. Figure 2.3 illustrates how we can value the bond.

Figure 2.3 A 4-year 6% coupon bond yielding 5%: backward sweep

Instead of going back in time step-by-step to discount and cumulate cash flows, we can value the coupon bond as the sum of its parts. The coupon bond is a package of four zero-coupon bonds with face values of $60, $60, $60, and $1,060 and maturities 1, 2, 3, and 4 years, respectively. We do this by multiplying each coupon cash flow (face value of the component zero-coupon bond) by its appropriate discount factor. This factorization is illustrated in Figure 2.4.

Figure 2.4 A 4-year 6% coupon bond yielding 5%: sum of the parts

Also, we can factorize by separating the $1,000 par from the rest of the cash flows:

The two methods in Figures 2.3 and 2.4 – the backward step-by-step discounting sweep and the sum of the parts’ present values – are equivalent and always produce the same answer. Mathematically, it makes no difference whether we multiply n times in succession by one-period (forward) discount factors

or all at once by today's (spot) discount factors

as long as the rates are all equal.

One has to be a bit more careful, but the two methods are still identical when interest rates for different periods are not the same. We deal with the more complicated case of changing interest rates in Section 2.3 where we call forward zero rates and rn spot zero rates. We develop further the notion of forward rates in Chapter 3.

2.1.3 Amortizing Bond

An amortizing bond is also a multicash flow bond. Unlike the coupon bond which has an equal stream of coupon interest plus a one-time cash flow representing the return of par, the amortizing bond has a stream of equal cash flows, all of which can be thought of as partly coupon interest (on the remaining balance of the loan) and partly principal repayment. This distinction is somewhat artificial since the coupon interest of the coupon bond is generally not the same as the interest rate earned on the amount invested in the bond. The real difference between the coupon bond and the amortizing bond is the pattern of cash flows. The amortizing bond's cash flows are all identical, and there is no large “balloon” payment at maturity.

Figure 2.5 illustrates a 4-year amortizing loan of $1,035.46 yielding a 5% interest rate. At that rate, the lender receives $292.01 each period. (Think of a refrigerator priced at $1,035.46 financed with a 4-year installment loan at 5%.) We value it using the sum-of-the-parts method.

Figure 2.5 A 4-year amortizing loan yielding 5%: sum of the parts

2.1.4 Floating Rate Bond

The last type of bond, the floating rate bond, is also a multiple-cash flow bond. Similarly to the coupon bond, it has a stream of coupon cash flows and a large balloon cash flow equal to the par value at maturity. However, the coupon cash flows are not known up front, but are set one period in advance: next coupon payment is set today, the coupon paid 2 years from today is set 1 year from today, the coupon paid 3 years from today is set 2 years from today, etc. The idea is to reset the interest rate, as on a revolving loan, to a new “fair” rate each year, rather than holding it constant throughout of the life of the bond.

Since the cash flows are not known in advance (we use a tilde to denote a quantity unknown today), it...