# Bond Pricing

We have built all the tools needed to approach the topic that is at the center of any discussion of credit and the activity of a treasury: debt. We have seen how to generate and discount future cash flows, we have seen how a choice of discounting is highly sensitive to the credit environment, and we have explicitly discussed credit. It is now time to use this knowledge to observe and price debt instruments.

In order to build a self-contained narrative we shall begin with an introduction to the basic concepts surrounding a bond. We shall then move on to the very important issue of trying to isolate the credit component of a bond in a more or less explicit way; we shall present the concepts of benchmarks, asset swaps (introduced here and revisited in the following chapters); and an analysis of the relationship between bonds and credit default swaps. We shall conclude with a section on how to price distressed and/or highly illiquid bonds and one where this final topic is presented through a numerical example.

## WHAT IS A BOND?

We have defined a bond as a way for an entity to raise capital without relinquishing control. Bonds are among the oldest financial instruments and among the first types of securitization,[1] a way of turning the scattered revenues of a government or a corporation into a well-defined and tradable instrument. An entity forecasts a fairly regular set of revenues (from sales for a company or from taxes and investments for a government) but needs an immediate and large cash amount: on the back of its revenues it will issue a bond in which it promises to return the amount at maturity and to pay interest at regular dates. The regularity of the entity's revenues should ensure the regular payment of coupons.

An investor will decide to buy a certain amount of this bond (effectively lending money to the entity) and the price for this bond will be driven by the investor's trust in the entity's abilities to meet its obligations. The price at a time t of a bond B with unit principal is defined as

(5.1)

where we remind ourselves that Ci are fixed or variable coupons, Di are the discount factors, and are the day count fractions.

Let us state first what might puzzle the mathematically minded reader who appreciates the form given by the above equation: a bond price is just a number. The prices of bonds are driven by people reacting to different stimuli but who almost never think of those prices as given by the neat sum of cash flows described above. This is not saying that traders trade in an imprecise fashion, it simply means that, through instinct and experience, the bond price is enough to mean everything to them. A musical comparison would be with those musicians (rare but who apparently can count Louis Armstrong in their midst) who cannot read music and yet are fully at ease with scales and (basic) musical theory. Of course, siding with the mathematically minded among us, we need to admit that this can only take the practitioner up to a point: the same way one cannot write the Art of the Fugue without reading music, one cannot go much beyond bond trading by simply thinking in terms of bond prices.

Derivatives are priced through models that require calibration. To calibrate means to take an observed (i.e., traded) quantity and use it to arrive at another, often more than one, hidden variable in the model: these model variables are implied from market data. The trader only sees the market data and it is enough for him, the quant sees the model variables as well.

Let us consider, for example, the CDS definition given by Equation 3.17: a quant, particularly one fairly new at his job, is often baffled by the ease with which traders think of CDS spreads, their movement, sensitivities to their movement, and so on. This is because, for the quant, what counts is the left-hand side of Equation 3.17, with its integrals, survival probabilities, and optimization. The trader's only concern is the fact that if he sees the value of C go up, the credit standing of the entity in question is deteriorating; if it goes down it is the opposite.

The same applies to bond prices: the quant sees a long list of coupons, forward rates (if it is a variable coupon bond), and discount factors; the trader sees the price. We shall see that this is not entirely true when dealing with a bond's yield, but the main point of this aside is a warning to the young mathematically minded practitioner not to freeze in front of a perceived great complexity. A final analogy will be with language. When learning a foreign language written in a different alphabet, at first we are forced to read words by going through each letter. We are amazed at the speed with which native speakers read, thinking that they must read the letters at great speed. We forget that with practice no one reads a word by scanning the letters, we read words by reaching stored images in our brain of the word itself. Although a bond is made of coupons and discount factors, the seasoned practitioner does not scan and price each of those, he simply concentrates on the overall image, the price.

The relationship between the different elements of Equation 5.1 is, however, very much real and crucial to understand for those of us who are interested in the meaning of the deeper factors that drive an entity's debt. These factors are often implicit and therefore need to be implied from the full definition of a bond. In order to do so let us introduce a few important concepts about bonds. These concepts are presented at an introductory level to render this chapter self-sufficient and to present them in the same formalism and with the same spirit of the remaining part of the book. For a more thorough and extensive discussion of bond fundamentals, the reader is directed toward Fabozzi [38] or Smith [76].

• [1] The interested reader is directed toward Niall Ferguson's The House of Rothschild for a riveting narration of what bond trading was in truly illiquid markets.