# Yield

In the previous section we said that the price of a bond at issue date is always 100: this is not entirely true, however. Since it is very close to par let us continue to consider it as a fix point. What makes bond pricing very interesting is that in a very simple way it allows the user to experience what implying a value means; moreover, since the variables entering the calculation are all roughly equally important, there is a true sense of choice in what we consider a given and what we want to imply.

Let us be more specific by using an example from a different financial field. In the case of an equity option, even assuming that price and volatility are equivalent in importance (which is not the case since a trader will always think in terms of volatility but a quant in terms of price), there are many other variables which enter into a fairly complex equation (Black and Scholes formula) in order to obtain either price or volatility. Interest rates (in the equity world) are always inputs; one would never think of using Black and Scholes' formula to imply them. On the contrary, Equation 5.1 is of a simplicity that anyone with a grasp of arithmetic can appreciate. In addition, we shall see how all its inputs are more or less equally important.

If at issue the bond price is, if not fixed, very close to a fixed value, if the coupon is fixed (we are considering for the moment the case of a fixed rate bond) then something else has to give way for the calculation to be correct. Indeed this is the case and our attention needs to focus on the discount factors.

We have assumed that *D*i in Equation 5.1 are the same as the discount factors we have seen up to this point, values implied from interest rate instruments such as deposit, FRAs, swaps, and even overnight rates. This is no longer the case since those instruments implied two specific types of risk. If the trade was collateralized, the implied risk rate was the overnight risk, otherwise, being those types of trades between market participants, it implied a level of risk of type LIBOR.

A bond is not collateralized (it would defeat the purpose of raising cash) and the issuance of a bond is not limited to financial institutions. This means that the level of risk is specific to the institution issuing the bond and this means that Di is no longer the same as the discount factors we have seen up to this point.

Let us then rewrite Equation 5.1 in the following way

(5.2)

where is a new variable. This notation is very different from the one found in the literature: it has its drawbacks and we shall soon converge to the more usual way of discounting bond cash flows. The wish here is to put discounting as far as bonds are concerned in the same mathematical form as discounting in the context of swaps, and discounting and survival probabilities in the context of CDSs. If this notation seems awkward now, when discussing the comparison between the debt and income of a financial institution, hopefully the reader will see how it has significant conceptual advantages.

We can consider in Equation 5.2 as a correction applied to the discount factor we are familiar with. This correction reflects the credit standing of the issuer.

Let us consider the case of two issuers, A and B, who on the same day issue a bond with maturity in five years' time. Let us consider a situation here similar to the one we have mentioned about Germany and another Euro country: A is the best issuer in one currency and B is one that has a considerably worse credit standing than A.

At issue date, A issues at 100.115 a five-year bond paying a 1.25% annual fixed coupon and B issues at 100.087 a bond with identical maturity and a 3.00% annual coupon. Table 5.1 shows the details of each bond. The first column displays the discount factor *D*i obtained through market instruments as discussed in Chapter 2, the third and fifth columns show the cash flows, that is, the coupon each year and the principal payment at maturity, and the second and fourth columns show the correction for A and B respectively. The correction for issuer A is equal to one: this means that the average rate (let us say the five-year swap rate as a proxy) used to build the

**TABLE 5.1** Cash flows of two similar bonds issued by two issuers with different credit standings.

A |
for A |
Cash flows for A |
for B |
Cash flows for B |

0.99560 |
1 |
1.25 |
0.98275 |
3.00 |

0.98974 |
1 |
1.25 |
0.96585 |
3.00 |

0.98069 |
1 |
1.25 |
0.94924 |
3.00 |

0.96430 |
1 |
1.25 |
0.93291 |
3.00 |

0.94027 |
1 |
100.25 |
0.91683 |
103.00 |

100.115 |
100.087 |

discount factors must be very close to 1.25%, the coupon value.^{[1]} The issue price of bond A,, is equal to 100.115 to make it realistic: had we insisted on exactly 100, the correctionwould have been slightly different from 1. The price at issue of bond B, , is also close to par but the corrections to the discount factors are very much different from 1.

We have implied the values of the corrections to the discount factors used in B's case by keeping both the issue price and the coupon fixed. The issue price is usually reached through an auction-type sale. There is an announcement of an imminent debt issuance on the part of an entity; the bond has a certain coupon and investors subscribe to the sale. The interest (or lack thereof) drives the issue price. This means that in real life things happen in a slightly different order. The corrections to the discount factors are driven by a market feeling for the entity's credit. The entity itself then sets, *before the auction,* the coupon so that, taking into account the market's feeling, the price is going to be roughly par.

Because the coupon usually takes values no more granular than a quarter of a percentage and because at the auction itself, market feeling can change, the price of the bond cannot be exactly par at issuance. This does not change the fact that an investor at maturity receives the par value. When the price is greater than par, the bond is said to be sold at a *premium;* when it is lower, it is sold at a *discount.* From this little exercise we hope we have shown what was considered interesting as far as bond pricing was concerned: we have played with three elements, the bond price, the coupon, and the corrections and we have shown how, at different moments, some can be kept fixed and some others can be implied.

It is probably time to stop calling a correction and move toward a notation more in line with the literature. In Section 2.2.1 and elsewhere where we discussed zero rates, we have shown the relationship between a discount factor and a simple rate. Using a similar approach, let us write

(5.3)

substituting into Equation 5.2 we obtain

(5.4)

where we have introduced *Y*, the *yield* of the bond, or to be more precise the *yield to maturity* of the bond since the calculation assumes that we intend to hold the bond up to maturity. (When pricing bonds, details are very important but some are beyond the scope of this book. The accounting for time, which we have simply expressed as *T*i – *t,* presents many nuances based on conventions, day count, holiday calendars, etc. The interested reader is directed toward Smith [76] for an in-depth treatment.)

It is important to note that the yield is one single number. Someone, admittedly misled by our previous formalism, might have been inclined to draw a parallel with discount factors or survival probabilities and write something along the lines of

(5.5)

or even

(5.6)

(note the suffix as opposed to Equation 5.3). This would have been radically different from the concept of yield, which draws its strength exactly from the fact that it is one single number (in nature it is similar to the concept of internal rate of return). Why is this important?

We have stressed at length how, once we set ourselves the target of par value, the higher the coupon the larger the credit-driven correction needs to be and vice versa. Conversely, fixing the coupon price, different levels

of corrections will lead to different bond prices. To think in terms of , however, is not easy.

In its simplicity a bond yield allows us a very quick calculation (or at least a feel for the magnitude): knowing either only the bond price but not the coupon or only the coupon but not the bond price, the yield lets us estimate the other very quickly. For example, if a bond trades below par and the yield is *y,* then we know that the coupon *c* must be such that *c < y.* Conversely, if we know that a bond pays a coupon C and I know that the yield is such that *C* > *Y*, we also know that the bond trades above par.

Let us stress the importance of the concept of yield by looking at the word itself. A bond can be roughly considered to be made of two parts, the principal and the coupon. The principal is the amount of money the investor lends to the borrower and is the main component of the deal, the coupon payments can be seen as the compensation the investor requires for delaying the consumption of the principal.^{[2]}

We have seen that the borrower sets the coupon by making sure that, once the bond is discounted and taking into account the market's perception of the borrower's own credit risk, the price is more or less par. This means that the coupon is the element taking care of the borrower's credit. Bonds, however, can be very long dated instruments and between the time the coupon is set and a subsequent time, the credit standing of the borrower might change, making the coupon value irrelevant as a credit signal. Moreover, irrespective of the price paid for the bond, at maturity the investor receives the full principal amount, the unit of measure being the principal amount. Here is where the yield plays a crucial role in telling how much the investment is really yielding. If an investor pays 95 for a bond with a 5% (of 100, of course) annual coupon, then the investment *yields* more than simply 5%.

For the most liquid bonds the yield plays such an important role that it can be traded alongside the bond price (not separately, of course, as the trader chooses to quote one or the other).

- [1] If we pay the same rate we use to discount, we must be at par or, to see it in the same light as this exercise, if we are at par, the rate we use to discount must be the rate we are paying.
- [2] This might seem a very roundabout, economics-driven way of defining coupon payments. It is meant to be general and linkable to the time value of money; other interpretations can be thought of.