Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

More on negative swap spreads

The universal feedback that I got on my last
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

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
  3. The 30 year bond yield is the fixed rate at which the US can
  4. The T-bill yield must be the floating rate at which the US can

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
  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

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

While this example is theoretically possible it is clearly
unrealistic. The purpose of my previous
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


3 responses to “More on negative swap spreads

  1. mcooganj September 23, 2009 at 5:04 pm

    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.

  2. mcooganj September 23, 2009 at 5:08 pm

    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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