More on negative swap spreads

The universal feedback that I got on my last post on this subject was that it was very difficult to understand. So let me try again.

At the outset, let me state that in my view the negative swap spread is a result of market dislocation; I do not even for a moment believe that it is really a rational market outcome. Yet, some people are making the argument that the negative spread is rational and can be explained in terms of default risk. I am therefore, trying to analyze (and hopefully) disprove this claim; mere hand waving is not enough.

Specifically, the claim being made is that the fixed leg of the swap is less risky than the 30 year bond because there is no principal payment at the end. So I begin by making the extreme assumption that the 30 year bond can default, but all the promised payments in the swap will be paid/received without default even if the government and one or more Libor rated banks default.

My initial thinking was that:

1. Libor is the floating rate at which a Libor rated bank can borrow
2. The swap rate must be the fixed rate at which such a bank can borrow
3. The 30 year bond yield is the fixed rate at which the US can borrow
4. The T-bill yield must be the floating rate at which the US can borrow

If all this is true, then by assuming that the T-bill yield is always less than Libor, it would appear to follow that the bond yield must be less than the swap rate.

Unfortunately, this simple minded analysis is inadequate because it assumes that interest rate risk and default risk can be nicely separated from each other and do not interact. The interest rate risk is reflected in the spread between Libor and the swap rate (a rising yield curve) and also in the spread between T-bills and the long bond (again, a rising yield curve). The default risk is reflected in the spread between T-bills and Libor and also the spread between the long bond yield and the swap rate. The world would be so simple if these two risks were orthogonal to each other and did not come together in crazy ways.

To understand this interaction, suppose that on the date of default somebody makes good the default loss to us by paying us the difference the par value of the bond and its recovery value. The default loss is therefore eliminated. Does this mean that there is no loss at all due to default? No, we are now left with the par value of the bond in our hands, but that is not the same thing as receiving the remaining coupons and redemption value of the bond. If we try to invest the par value of the bond, we may not be able to earn the old coupon rate if interest rates have fallen.

A default in a low interest rate scenario is in some ways similar to a bond being called. In fact, a default with 100% recovery is completely identical to a call. Conversely, a default in a high interest rate environment has some similarities to a put; and the similarity becomes an equality if recovery is 100%. Therefore, in addition to the default risk, we need to consider the value of the implicit call or put that takes place when the bond defaults.

The situation that I envisaged in my previous post was that if the US government defaults only in a low interest rate environment, its yield must include a premium not only for default losses but also a premium for its implicit callability. The swap rate will be the yield on a non callable bond, because the swap continues even if one or more Libor rated banks default. I am assuming that the risk of the swap counterparty defaulting is taken care of by sufficient collateral. If the yield sweetener required for the implicit callability of the US Treasury outweighs the extra default premium (the TED spread) embedded in Libor, it is possible for the Treasury yield to exceed the swap rate. I emphasize that I do not consider this likely, but it is a theoretical possibility.

To demonstrate this theoretical possibility, I now present an admittedly unrealistic numerical example where this happens. I assume a default risk on US Treasury of about 15 basis points annually while Libor contains 30 basis points of default risk embedded in it. From a pure credit risk point of view, the Libor rated bank is riskier than the US, but in my extremely artificial model, the 30 year swap rate is only 4.06% while the 30 year US Treasury yield is 4.27% (roughly similar to early September numbers). This happens because in this toy model, Treasury default is perfectly correlated with interest rates and amounts to callability of the bond. In this model, the yield on a hypothetical default free 30 year non callable bond is only 3.76% while the yield on a default free 30 year bond callable after 10 years is 4.08%. This means that the hypothetical default free callable yields more than the defaultable non callable swap. The defaultable Treasury has to yield more than the default free 30 year callable to compensate for default risk.

The precise model that yields the above numbers is as follows. The US Treasury defaults with 10% probability exactly at the end of 10 years with a recovery of 55%. This corresponds to an expected default loss of 4.5% or 15 basis points annualized over the 30 year life of the bond (in present value terms, the annualized default loss is obviously slightly different). The default free term structure over the first 10 years is roughly similar to the actual US Treasury yield curve in early September. The only two numbers we need are the 10 year zero yield (3.59%) and the 10 year par bond yield (3.45%).

At the end of 10 years, there are two possibilities:

1. The US government defaults and the risk free rate remains constant at 0% (zero) over the next 20 years. The probability of this is 10%.
2. The US government does not default and the risk free rate remains constant at 4.75% over the next 20 years. The probability of this is 90%.

Note for the finance experts: all probabilities above are risk neutral probabilities.

In this model default is perfectly correlated with interest rates and a defaultable bond with 100% recovery would be the same as a default free callable bond. This allows us to decompose the 51 basis point spread (4.27% – 3.76%) of the US bond over a default free non callable into two components: a callability component of 32 basis points (4.08% – 3.76%) and a default loss component of 19 basis points (4.27% – 4.08%). The swap is non callable and its entire spread over the default free non callable bond of 30 basis points (4.06% – 3.76%) is due to default risk. This default loss spread is 11 basis points more than that embedded in US Treasury indicating that it has higher default risk. This 11 basis points can be interpreted as the average implied TED spread over the entire period.

While this example is theoretically possible it is clearly unrealistic. The purpose of my previous post was to prove that under realistic assumptions, it is not possible for the US Treasury yield to exceed the swap rate even if we assume that the swap payments will continue without default even after Treasury has defaulted. But that argument is necessarily abstract and complex.

Posted at 8:59 pm IST on Tue, 22 Sep 2009         permanent link

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