In Figure 5.1 between the spread to a reference swap rate and the z-spread we see the asset swap spread quoted. In order to appreciate its importance we need to show the basics of an asset swap structure.

As represented in Figure 5.2 an asset swap consists of a structure where an investor (the asset swap buyer) holding a bond transfers the bond's coupon onto another party (the asset swap seller), who in returns pays the investor LIBOR plus or minus a spread, the asset swap spread.

One could consider an asset swap as a way, on the bond holder's part, to achieve two things: to monetize the bond's credit risk and to do this relative to a standard benchmark such as a LIBOR. The ownership of the bond remains with the bond holder (the asset swap buyer); in case of default the bond holder will bear the brunt of it settling for the recovery rate of the bond. In case of default the swap does not terminate automatically, meaning the bond holder is still liable for the swap coupon payments unless the swap is exited (which usually is done at terms that are suboptimal for the asset swap buyer in these situations^{[1]}). The asset swap spread is meant to offer the bond holder a way to hedge the credit risk of the bond. How is the asset swap spread calculated in practice?

FIGURE 6.3 A detailed representation of a par asset swap, where P is the bond price.

Before proceeding let us elaborate on our simple picture given by Figure 5.2 where we specify that it shows a par asset swap at inception. This means that the swap was entered into at the same time the bond was issued and both instruments were worth par. While, we shall see in great detail later, a treasury entering into an asset swap structure will always do so at inception and usually at par, let us for the moment maintain a general tone and assume that we are dealing with the situation of an investor who purchases a (fixed rate, for simplicity) bond not at inception and wishes to swap it for a stream of floating payments. The investor has two options, either to enter a par asset swap as shown in Figure 5.3 or a market or proceeds asset swap as shown in Figure 5.4.

FIGURE 6.4 A detailed representation of a market (or proceeds) asset swap, where P is the bond price.

In Figure 5.3 we have assumed that the bond the investor is swapping is worth less than par. In this situation, one of a par asset swap, the investor needs to pay the asset swap seller up-front the difference between par value and the current bond price. This is because otherwise the investor would begin the swap with a free mark-to-market gain. One could think of this in terms of a fictitious principal exchange: if we were to exchange par principals, the seller would offer the full principal and the buyer would need to do that as well, hence the difference between the bond price and par. During the life of the swap the buyer turns over the coupons C (on a full principal) to the seller and the seller pays a spread over LIBOR to the buyer. At maturity each would pay the full final cash flow and the buyer would also receive the principal from the bond issuer. Now that we have seen the picture in full, how do we calculate the asset swap spread? The situation when the bond is currently worth P, seen from the point of view of the investor, can be expressed mathematically as

In the above on the right-hand side we have the purchase of the bond at a price P and an income stream of coupons discounted on a curve adjusted for the credit risk of the issuer. On the left-hand side we have the up-front payment and, for the investor, an income stream of LIBOR payments plus a spread sa and a cost stream of coupons all discounted on a LIBOR-driven (or OIS-driven as shown in Chapter 2) curve. The asset swap spread sa is calculated such that at inception the above structure is fair. In the above we have assumed only one currency for the sake of simplicity and we have assumed that the frequency of the bond is the same as the floating leg payments. We now see how useful it is to use the terminology introduced in Section 5.2.2 in which we use a credit correction to the discount factor. This means that on the right-hand side of Equation 5.11 we do not need to concern ourselves for the moment with how we discount the bond, how we get to the correct yield, and so on. What is crucial is that the same fixed coupon Cj is discounted in two different ways on two sides of Equation 5.11.

In Figure 5.4 we describe the situation of a market (or proceeds) asset swap. The investor (or asset swap buyer) has bought a bond at a price P, which is less than the full par value, and instead of entering into a swap with principal 1 he enters into a swap with principal P. During the life of the swap the investor pays the coupons from the bond and receives a set of floating payments on a P principal. However, the coupons and, crucially, the final payment, since they need to match the bond, refer to a full principal which means that the asset swap buyer will pay 1 at maturity to the asset swap seller who will need to increase the natural final payment of the swap, which would have been P, by an amount (1 – P). Mathematically we can express this, again from the point of view of the investor, as

(5.11)

(5.12)

The asset swap spread is again the spread that renders the whole structure above fair at inception. For other types of asset swaps and an interesting discussion on the sensitivities of asset swaps, one should see O'Kane [66]. In Equation 5.11 we have highlighted the initial up-front payment that corrects for the fact that the bond price is not par. Previously, in Equation 5.12 we have highlighted the last additional payment, and in order to do so we have included the last principal exchange cash flows in the swap on the left-hand side. As odd as it might seem, this is to show that, whereas in the par asset swap the principal is the same on both swap legs (the principal exchanges are nonexistent since it is a single currency swap), in the market asset swap it isn't and this is because the coupon leg of the swap always has to mirror the bond. Perhaps it is useful to stress it again to avoid confusion. There isn't a real principal exchange at the end, only the additional payment on the asset swap seller side. We have written it in this form in Equation 5.12 for didactic purposes only.

Of the spreads that can be used to characterize the credit risk of a bond, benchmarks, swap spreads, and z-spreads are useful indicators to compare different issuers, even across different types of maturities. They are, however, not useful calculation tools and certainly they are not trading tools. No trader would trade a bond using the reference spreads to swap rates, benchmark bonds, or z-spreads (we shall see later, however, how they can be useful to arrive to tradable values for illiquid bonds).

Asset swap spreads, on the other hand, are not useful credit indicators because they are driven by the bond price, which in itself is driven by many factors, credit or noncredit related. The spread can be useful to compare two issuers only if we compare two bonds trading roughly at the same price and with similar issue and maturity dates. (For this precise purpose, on the other hand, an asset swap is an extremely useful tool and probably the only one rigorous enough.) However, asset swap spreads are real calculation tools and are tradable values, that is, it is a number a trader would quote. They are particularly important for our discussion because, as we shall see later, asset swap structures constitute the essence of funding and treasury operations. The funding level of an institution is in essence the asset swap at inception of a recently issued bond.

We shall conclude our analysis of bond credit with a look at the relationship between bonds and credit default swaps.

[1] This is true in general; however, as we shall see later, Treasuries use asset swaps in the opposite direction, that is, they issue a bond and then they swap the liability. In these situations there is a clear linkage between bond and swap, and usually when the bond terminates so does the swap (but not vice versa).

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