# Constructing a CDS-Implied Credit Framework for Bond Pricing

After discussing the credit element of a bond price, the reader might be tempted to wonder, given that credit default swaps were created to offer protection against bond defaults, can we leverage the information given by a CDS rate in order to price a bond? In principle this would be possible, but there are a few important caveats to consider.

Let us first take an intuitive step: we are going to obtain the price *B*t of a bond by multiplying each cash flow by the survival probability of the issuer in a way similar to the one we used to fair value loans in Section 3.3.1. In this framework a bond with principal *N* will have a price *B*t given by

(5.13)

where we have simplified the discounting by setting with *r*i some continuously compounded rate (very similar in nature to a zero rate). We have seen in Section 4.2.3 (and prove in Appendix B) that one could approximate the probability of default by dividing the CDS rate by one minus recovery, that is,

In turn, by using a simple binomial expansion one can show that, at least for short maturities,

Combining the above two equations we obtain

(5.14)

(Note that we have written *hj* instead of this is because, in our approximation, we assume it to be an annualized rate like the CDS rate on the other side of the equation.)

Writing the survival probability explicitly in Equation 5.13 and using as hazard rate the value found in Equation 5.14, we obtain for *B*t

(5.15)

We also know, using Equation 5.10 with a rate continuously compounded, that the price of a bond can be expressed by adding a z-spread *z* to an interest rate component and therefore write *B*t as

(5.16)

From Equation 5.15 and 5.16 we see that a CDS rate is (roughly because of some approximations) equivalent to a z-spread in the special case of a zero recovery bond. The situation of a zero recovery bond, however, is an idealized one and in practice (as shown in great detail by Berd et al. [12] and Pedersen [68]) things are more interesting.

From the above we see that the shortcut taken by some of simply treating the CDS rate as a z-spread and using it to build a correction to the discount factor is fundamentally flawed. To divide the CDS rate by one minus recovery (essentially using Equation 5.15) is an improvement. A more sophisticated approach would be to calibrate a set of survival probabilities *S*i to CDS rates as shown in Section 3.2.2 and write the bond price *B*t taking into account recovery, that is,

(5.17)

where we have returned to writing discount factors as *D*i, the more general form.^{[1]} Equation 5.17 states that we have a stream of cash flows including a final payment that are all contingent on the survival of the issuer plus a payment of one minus recovery in case of default.

The approach given by Equation 5.17 is more sophisticated, but sometimes it can misrepresent reality. However, at this point we need to state our objective clearly. To calibrate or to imply (as we will discuss further in Section 5.4.1) model variables means using a model to reprice a market variable: it is crucial that one is observable and the other is not.

Although the above statement is correct in all situations, if we assume that whenever something is not observable we are using a model to find it, let us rephrase things slightly differently. In finance we often use one piece of market information to find another piece of information/variable. Almost by definition, if we are trying to find the latter, this is unknown. There are some situations, however, where this is not the case and we are facing what at first appears to be an inconsistency.

When dealing with bonds and CDSs we might be in one of three situations: we have bond prices, easily available but not CDS rates, we have CDS rates easily available but not bond prices, or both are easily available.

In the first situation, Equation 5.17 will be a calibration tool rather than a bond pricing tool. Since we do not have CDS rates easily available we cannot arrive at Equation 5.17 with an already calibrated set of survival probabilities obtained as shown in Section 3.2.2. Instead we would use it to obtain *S*i. Of course in order to build a term structure of survival probabilities we would need an equation such as Equation 5.17 for several different maturities.

In the second situation (an absence of bond prices), we would use Equation 5.17 as a pricing tool and we would be satisfied with the value of *B*t. Note that in this statement we are using an “absence of evidence is evidence of absence” approach: if we cannot show that the bond price obtained through the use of Equation 5.17 is wrong, then it must be correct. In this we are using a principle similar to the one that can be applied to the currency basis swap introduced in Section 2.3: if it is not traded we can assume it to be zero.

The mentioning of the currency basis (in the context of the calibration of *S*i to existing CDS quotes) is not casual. We could find ourselves in the third situation, one where we use Equation 5.17 and it does not lead to to the traded price *B*t. The discrepancy between the two is explained by the *bond-CDS basis.* We have mentioned before that in finance a basis is a way of quantifying and trading what appears to be an inconsistency. In this case the inconsistency is the fact that the survival probabilities implied from CDS rates do not describe the credit risk associated with a bond. We said in Section 3.1.1 that a CDS is a way of turning a risky bond into a riskless one. If we purchase a risky bond we have a large risky return: if we choose to use some of that return to purchase protection in the form of a CDS then we expect to have the same (lower) return as a riskless bond (e.g., a treasury bond). This is valid in theory but in practice we can find ourselves with a higher or lower return according to the sign of the bond-CDS basis. In this situation we need to write Equation 5.17 as

(5.18)

where the new survival probability *S,* is obtained (a variation of Equation 3.16) calibrating to

(5.19)

where, for simplicity, we have assumed the old quotation with only a running coupon C and where *CDS*basis can be either positive or negative.

Although the idea of calculating a bond-CDS basis (in itself a double calibration, of the basis and of the survival probabilities) might seem pointless when we have both CDS rates and bond prices, there are situations in which it could be useful. First, there could be a situation where the presence of both bond prices and CDS rates happens only for certain maturities: the basis found for those maturities can then be applied with some extrapolation to those maturities where we only have, say, CDS rates. In the same spirit, which we shall apply in Section 5.4.3, a basis found for a certain issuer can contain useful information that we might use to price similar bonds issued by other entities.

- [1] Some readers might be slightly annoyed by this continuous change in the discount factor formalism. As an apology we can only say that we find the general form
*D*i intellectually pure, based on the fact that everyone agrees on what the concept of present value is (but not on how to get there). We abandon it only when it is really necessary and try to revert to it as soon as possible.