# Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

## More on negative swap spreads

September 22, 2009

Posted by on 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:

- Libor is the floating rate at which a Libor rated bank can borrow
- The swap rate must be the fixed rate at which such a bank can

borrow - The 30 year bond yield is the fixed rate at which the US can

borrow - 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:

- 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%. - 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.

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i think what you’re saying above is that back end risk is why LIBOR bank bonds price swap ++, and why 3/6 floating basis is +ve?

if that’s the case, i think you’ve assumed away the problem – so you’re not really trying to explain it. In your example, balance sheets are infinite. that’s not currently a decent working assumption.

it’s not the risk associated with the back end payment on the bond that makes bond yields higher than swap yields — it’s the value of the cash upfront to buy the bond.

a 30yr swap at market is an off balance sheet derivative with zero net present value on day 1. sure it costs margin, but that’s a lot less balance sheet intensive than a 30yr bond.

As balance sheets open further, their value will fall, and swap spreads will normalise – as it will become worth while asset swapping 30yr bonds.

another way of looking at this is that asset swapping a Bond yields 20bps of carry per year. there have been much better uses of balance sheet than a 1.5bps per month carry trade – because the market has been so dislocated.

as balance sheet constraints ease, and markets heal (so the easy goes) asset swapping bonds gets more worthwhile.

right now, it’s not interesting / worth it

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