 # EXPRESSING CREDIT EXPLICITLY WHEN PRICING A BOND

In Sections 5.2.1 and 5.2.2 we spoke at great length of the role played by the credit standing of the borrower when it comes to pricing a bond. In particular, we have seen how the concept of yield plays a fundamental role by simply and directly combining information about the coupon of a bond and the worth (hence the implicit credit risk) of the investment.

There are, however, situations in which we do not want to use such an immediate tool and we wish to isolate the credit element of bond price. We shall introduce some of the most useful ones.

In Equations 5.5 and 5.6 we have shown how not to calculate a bond yield. Those equations nonetheless are examples of other attempts to separate the interest rate component from the credit component in the discount factor applied to bond cash flows.

We have repeated many times, beginning in Chapter 2, that when we deal with a set of payments that includes a final principal, and if we discount with the same rate we use for coupon payments, the value of the leg is more or less par. This means that if we have a par leg paying a higher coupon rate than the one used for discounting or if we have a leg paying a similar coupon but which prices to less than par, there must be some further parameter involved in discounting. An intuitive way of visualizing this extra element is to think of a spread we add on to the normal discount rate (by normal we mean the one implied from interest rate instruments).

We would then write the bond value as (5.10)

where Ri is some interest rate and b is some spread we add to it. For the avoidance of doubt, in the absence of b, we would imply from Ri the usual discount factors Di. In the way the discounting is written in Equation 5.10, we have separated an interest rate component from a credit component. The credit component should be technically defined as a non-interest- rate component in the sense that it might include factors (such as liquidity for example, see Schwarz ) that are beyond credit; however, since credit risk is the main driver of b we can safely call it a credit component. One could even claim that nowadays to suggest the existence of a pure interest rate component (with its risk-free connotation) does not make much sense since, one of the main arguments of this book, credit is everywhere. To be more precise then, we could call Ri the component that could be implied from interest rate instruments and b the other, mainly credit-driven, component. Being a rather awkward set of definitions, we will remain with the former being the interest rate and the latter the credit component.

Once we have separated the two components we can decide what rate we use as reference point, that is, what the rate Ri is that we want to measure the credit risk of our bond relative to. The choice of reference points is usually driven by the habits of the trader using it. Some options are shown highlighted in Figure 5.1 where we look at the quote for a French government bond paying a 3.75% coupon and expiring on January 12, 2012.

The most important reference point is the benchmark, that is, a benchmark bond to which the bond we are considering is compared to. This is usually the most liquid and least risky bond in a certain currency. For example, in the case of the French government bond shown in Figure 5.1, the benchmark bond is the German three-month government bond.

The benchmark spread of x bps states that the bond we are considering yields x bps more (hence is riskier) than the benchmark. The number is usually always positive since, as we said, the benchmark tends to be the least risky bond available in that currency. In the example shown in Figure 5.1, the French government bond yields 30.9 bps more than the three-month German government bond. All the bonds issued by Euro area countries are measured against German treasuries (government bonds are often referred to as treasuries). In late 2011 the spread varied from a few tens of basis FIGURE 5.1 A sample quote for a French government bond with the different benchmarks highlighted. Source: Thomson Reuters Eikon.

points, as in the case of France, to a few thousand basis points, as in the case of Portugal and Greece.

If we want to apply the benchmark spread information we take the yield Y of the benchmark and we set Ri = Y (note the suffix: we set all Ri equal to the yield) in Equation 5.10 and then set b in the same equation equal to the benchmark spread. In light of what we have just said about interest rate and credit components, it might seem a little odd to set Ri equal to Y. Didn't we just say that a yield also contains credit information and Ri shouldn't? The question is certainly valid but we are making the assumption that the benchmark is so devoid of credit risk that its yield is almost solely driven by interest rate moves.

The next reference point of note is the spread given in comparison to a reference swap rate. In this case we take a swap rate–in Figure 5.1 the value shown is 1.528% and we are told, not surprisingly, that it is the local EUR interest rate swap rate–and we add the swap spread, which we are also told is – 95.7 bps. This combined value is then used as a yield in our calculations.

In the case of a government bond of a country with a good credit standing such as France, we are not surprised to see that the spread is negative. Let us remind ourselves that a swap rate is a LIBOR-driven value, that is, it carries a risk of borrowing of the same order of magnitude of an investment bank. One would assume a financially sound government to have a lower level of risk. Let us do a little check to see if all these numbers make sense.

All spreads are relative measures, however, we also have an absolute quote, the reference swap rate. If we take the reference swap rate and add the (negative) swap spread we are left with 0.571%; if from this value we subtract the benchmark spread of 30.9 bps we have 26.2 bps. What should this value represent? It should give us the yield of the German treasury, which, with an actual value of 25 bps (not shown in the figure), is close to the implied value we have just calculated.

The benchmark spread and the spread relative to the swap rate are very important because they give each a sense of relative measure compared to two very different worlds: the almost riskless world of treasuries and the risky world of interdealers borrowing.

The next spread and probably one of the most commonly used is the z-spread. This spread is the value that one would add to the zero rate of the corresponding maturity. This means that in Equation 5.10 R, is the zero rate corresponding to the cash flow at time Ti and b is the z-spread.

We have mentioned before in Section 2.4.3 that a zero rate is not a traded instrument and as a consequence, although it has the same structure as a cash deposit, it is not an official quantity as to whose value all market participants agree. However, because of its simplicity it is also a quantity everyone can easily grasp conceptually. If a bank A wants to borrow a certain amount of money, what is the simplest rate it can be charged? A rate fixed today payable every year, that is, the zero rate. Once we grasp this very simple concept and imagine the situation of a second borrower, a nonfinancial institution B, it is easy to see the extra cost B will be charged as a spread over the zero rate.

We have been surprisingly specific as to the nature of the business of A and B. Why? Although the zero rate is not a traded quantity, it is assumed that it is a quantity calculated after having built a curve according to the methodologies shown in Chapter 2. Since those methodologies involved the use of a great deal of LIBOR-driven market instruments, it is assumed that a zero rate would carry the intrinsic risk level of a financial institution. It is then understandable that a government such as France would have a negative z-spread since we would expect it to have a safer credit standing than the average institution relying on LIBOR levels of borrowing.

As one can see from Figure 5.1 there are other spreads shown in a standard bond quote. Some go beyond our scope and some, such as the relationship between bonds and CDS, will be treated later. The last of the most FIGURE 5.2 A schematic representation of a (par) asset swap at inception.

common reference spreads is the asset swap spread, and since it is central to our discourse on funding, we shall discuss it in a specially dedicated section.

•  The italics are due to the fact that, as we shall see throughout the rest of the chapter, to imply variables from one traded quantity, the bond price, is not very easy, which means that not everything can be precisely implied at the same time. Being people usually more interested in credit, the interest rate component is generally assumed rather than implied and used as a springboard toward implying the credit one.
•  As a reminder, we have defined the relation between the zero rate Ri and the discount factor Di as 