# Credit Default Swaps

Let us observe what happens in the case of a default. We have purchased a bond issued by ABC with unit principal paying a certain coupon and repaying the principal at maturity T. The bond issuer defaults (let us imagine that the event is clearly defined and accepted by all parties) and can no longer honor its obligation with respect to the bond we have purchased. Together with all the other creditors, we participate in the assessment of the assets of ABC to see what can be salvaged and distributed to us all according to the rules of liquidation. The fact that there is the chance of obtaining something at the end means that the bond after default is not worth zero, it is worth some value R. R is the expected recovery rate of the bond, the amount, as the name suggests, that can be assumed to be recovered from the default. The value of R is unknown in advance and there are many subtleties about it that we shall see later. Let us treat it, at the moment, simply as what we expect is left of the bond after default.

After purchasing the bond, in order to protect ourselves against the default of the issuer we decide to enter into a transaction with another counterparty, XYZ. In this contract, shown schematically in Figure 3.1, with

FIGURE 3.1 A schematic representation of a CDS contract with a) physical exchange of the bond and b) without.

First of all, let us point out that we are considering here the traditional structure of a CDS agreement in which the protection buyer delivers the bonds upon default, that mathematically to say that in case of default we obtain (1 – R) and to say that we exchange the bond for the principal with XYZ is identical. However, there is a subtle difference that should force us to remember that it is the latter that actually takes place. (1 – R) is a quantity that depends on the value of R if that is what XYZ gave us. Immediately we would be asking the question, what is R? The physical exchange between bond and principal makes us indifferent[2] to the actual value of R. We are

(3.3)

indifferent as far as the protection offered by the CDS is concerned; however, we can easily see from Equation 3.3 (where, similarly to a swap rate, C is traded at that value making both sides equal) that the premium we are going to be charged by XYZ is indeed dependent on R. In practice R is not a traded quantity[3] and is treated as an assumed quantity. In a normal environment it is a constant value for all issuers of a certain kind (e.g., 30% for all American corporate issuers); however, when one in particular is in danger of default, the actual value used is refined to a more, supposedly, precise value. We shall see, when dealing with illiquid bonds in Section 5.4.1, how this takes place. As far as the recovery value is concerned we shall return to this when discussing the fair value of loans and the different role it plays for normal financial institutions and development banks.

A second point is that, while in the definition of bonds and loans given by Equations 3.2 and 3.1, we have not shown their present value; in the case of a CDS we had to. In Equation 3.3 we have a traded quantity C (we shall see later that the survival probability is an implied quantity) that can only be traded if both parties agree on the value of each leg: this can only happen if we calculate the present value of the transaction. In the case of a bond, the value of the coupon is decided by the issuer and is not agreed between issuer and investor, which is why we could present it without present value. Of course the issuer will decide a coupon value giving the bond a value close to par, and the investor will decide what price to pay for that bond, but the coupon value is not agreed by the two parties. This point might seem odd at this stage, but not being central to the discussion of credit we can leave its clarification to a later stage.

It is not difficult to see how the riskier the issuer is the higher the premium C is going to be. In Figure 3.2 we see a plot of the term structures of credit default swap rates for Germany, France, and Korea. For example we see that if we want to buy protection for five years against the default of the German Republic we need to pay an annual premium of 98 bps. Korea is considered riskier by the market and for the same type of protection we need to pay an annual coupon of 162 bps. France is considered riskier still and if we want to buy protection for five years against the default of the French government we need to pay a premium of 205 bps.

We shall view this in greater detail when discussing bond pricing, however, it is already easy to see how, in theory, the combination of the possession of a bond with the purchase of protection basically turns an exposure

FIGURE 3.2 The credit default swap rate term structures for the republics of Germany, France, and Korea on January 18, 2012.

to credit risk into a risk-free situation. Before turning to a discussion of how this happens in practice through the concept of survival probability, let us end this section with a historical note centered around the combination of the possession we mentioned in the previous sentence.

CDSs were created to protect investors against the default of an issuer, meaning that if we purchase \$100M worth of ABC bonds, we (should) only need to purchase \$ 100M worth of protection. However, a CDS is a derivative instrument and, like all derivative instruments, it allows the owner to gain exposure to an underlying (in this case the credit risk of an issuer) without owning the underlying itself, a synthetic exposure. The same way we buy a CDS written on ABC's credit after having bought a bond issued by ABC, we could as easily buy the CDS on its own.[4]

Looking at Figure 3.2 we see that the five-year protection on France costs 205 bps. We sense that the market perception of the creditworthiness of France is deteriorating and without owning any French bond we purchase protection against France's default. In a week, the cost of protection against France's default has increased to, say, 225 bps, meaning that the 205 bps we paid is now worth 20 bps more. We sell protection against France to someone else and we make a profit. In all this we have not even touched a French bond. Moreover, imagining that our feeling was very strong, there was nothing stopping us from doing this trade on a principal far greater than the total amount of all French debt,[5] that is, applying considerable leverage to our position. Doesn't this negate the principle of protection, which is not very different from the one of insurance? It does but, again, it is a problem affecting all derivatives in general.

A derivative allows the user to break the economic link grounded in the real world between the derivative (the function) and the underlying (the variable). This opens the door to a potentially excessive use of leverage, by entering into a derivative contract well in excess of the natural size of the trade. This natural size of the trade could be, for example, the market capitalization of a company we write an equity option on. If we buy protection against the default of a borrower, it could be the total amount of debt issued by the borrower.

Let us imagine that the principal of our trades was a great amount, say, \$2BN (21 bps is a small number; if we want a large profit we need to multiply it by something large) and let us imagine that France defaults. On paper we should not worry: we are going to receive from one side the protection we need to pay on the other, but are we really sure of that? Some of these trades would be collateralized, but we have seen that some institutions do not pay collateral. The counterparty selling protection to us might in turn having bought it from someone else and so on until someone will have to find (1 – R) of \$2BN, which potentially could not be there.

• [1] Otherwise we would be facing the paradox in which the value of debt grows after default because of the high demand of all those needing to borrow it in order to deliver it as part of the CDS agreement.
• [2] In the case of physical exchange we are indifferent to the recovery rate R only if we buy a CDS to hedge our exposure to the credit of an issuer whose bond we possess. If we simply bought the CDS naked, that is, without a link to the actual bond, then we are not indifferent to the value of R.
• [3] There are recovery swaps trading the recovery itself but they are fairly rare.
• [4] This type of operation is sometimes referred to as naked in the sense that we bare the hedging link between derivative and underlying.
• [5] This is a bit unrealistic when it comes to the debt of a large developed country, but not in the case of a corporation or a smaller state.