Negative swap spread

The fact that the 30 year US dollar swap rate is lower than the
interest rate on the 30 year US treasury bond was till recently
something that only fixed income specialists worried about. Sure, the
Across the Curve blog has
been putting NEGATIVE in capital letters in each of his daily blog
posts on the swap spread for several months now, but the mainstream
financial media did not bother much about it. Last week, however,the
Financial Times carried a detailed story (“Negative 30-year rate swap
spread linger,” September 9, 2009) on the subject.

Under the current view that financial markets have normalized, the
negative swap spread is an embarrassment because it suggests that even
a year after Lehman, simple arbitrage trades are not happening because
of a paucity of the balance sheets required to put on the
trades. Alternative explanations are being sought for the phenomenon,
and the report states that “questions are being asked in the
market about the assumption governing whether a 30-year swap is
riskier than a 30-year bond.”

In this post (necessarily long and highly technical), I shall try
to examine this question. I shall initially assume that the interest
rate swaps have no counterparty risk because of high degree of
collaterilization. This is very different from asserting that the swap
rate is a risk free rate.

I shall assume that the Libor rate on the floating leg of the
interest rate swap is a rate that includes a default risk component. I
shall also assume that the default risk inherent in Libor is greater
than that of US Treasury. More precisely, I shall assume that the TED
spread (the excess of Libor over the T-Bill yield at the same
maturity) is expected to remain positive. I shall also assume that the
positive TED spread reflects the greater credit risk of Libor as
compared to the T-Bill. Before the crisis, it was fashionable in the
CDS market to assume that Libor and swap rates were risk free rates
and the TED spread was due to liquidity and tax effects. I believe
that this claim is untenable today.

Since banks are afloat today with huge government support, I think
it is reasonable to assume that the government is more credit worthy
than the banks. But I do not assume that the US government is risk free
either. It too can default, but the probability of this default is
lower than that of the banks.

Libor is the borrowing rate of a bank with what is often called a
“refreshed Libor rating.” On every day that Libor is
polled, only a sample of “sound” banks is
considered. Therefore, the default risk inherent in three-month Libor
is that of a bank defaulting in the next three months given that it
meets the Libor creditworthiness standard today. Libor exceeds a
hypothetical three month risk free rate by a compensation for this
possibility of default.

Assuming that the interest rate swap itself has no default risk,
the fixed rate payer should be willing to pay a fixed rate that
exceeds the risk free rate because what he receives on the floating
leg is higher than the risk free rate. He should also be willing to
pay more than he would on a swap in which the floating leg was the US
T-bill yield instead of Libor because I am assuming that the TED
spread (T-bill yield minus Libor) is expected to be positive. The
T-Bill yield itself exceeds a hypothetical risk free rate because of
the the possibility of default by the US government.

Unfortunately, even from all these assumptions, it does not follow
that the 30 year UST yield should be less than the 30 year swap rate
without some further assumptions that we will come to at the end. The
problem is that the interest rate swap is not terminated by the
default by one or more of the Libor rated banks or by the default of
the US government. Several banks may fail and Libor may still be
computed the next day based on the few banks that remain. The floating
rate payer on the swap would have to make floating leg payments at
this Libor rate, and the fixed rate payer would have to make fixed leg
payments at the fixed rate.

The holder of the 30 year bond however will not continue to receive
coupons if the US government has defaulted. To eliminate the default
risk of the US Treasury, we must consider a hypothetical asset swap on
the 30 year bond. Consider an asset swap in which (a) the owner of a
newly minted bond sells it to an asset swap buyer at par, (b) the
buyer agrees to make fixed rate payments at the coupon rate of the
bond, and (c) the seller agrees to make a floating rate payment at
Libor +/- a spread.

Assuming that the asset swap is risk free, the asset swap seller
now has a risk free stream of payments equal to the coupon of the 30
year UST bond. If it were true that the floating leg payment would be
equal to the T-bill yield, then we can immediately conclude that the
30 year bond must yield less than the fixed rate of the 30 year interest
rate swap. If not an arbitrageur would enter into an asset swap as a
seller and simultaneously enter into an interest rate swap as a fixed
rate payer. It would be left with two sources of profit from these two
swaps:

1. the fixed rate it receives on the asset swap would exceed the
   fixed rate that it pays on the interest rate swap because the 30 year
   bond yields more than the swap rate
2. the floating rate it pays on the asset swap (T-bill yield) would
   be less than what it pays in the interest rate swap (Libor) because
   the TED spread is expected to be positive.

If US Treasury were risk free, it is evident that the floating leg
would be equal to the T-Bill yield. We just add a notional exchange of
principal at the end (which simply cancels out). The fixed leg must be
worth par because it is economically the same as the newly minted 30
year Treasury (par) bond. Therefore the floating leg payment including
the notional payment must also be worth par, but this “floating
rate bond” can be worth par only if the floating rate is the
risk free rate which is the T-Bill yield.

This equivalence breaks down when US Treasury can default. To
understand this consider a modified asset swap which terminates
without any termination payments if and when US government
defaults. In this case, it is easy to see that the modified asset swap
floating leg must equal the T-Bill yield. The case where the US
government does not default has already been analyzed above, so
consider what happens if there is a default.

In this case, we add a notional exchange between the swap buyer and
the swap seller not of the principal value of the bond but of the
recovery value of the defaulted bond. With this notional payment
included, the fixed leg again is the same as the US treasury bond. It
must therefore be worth par because the Treasury bond is a par
bond. The floating leg must therefore also be worth par which means
that it (including the notional payment at default of the recovery
value) must be a par floater. But the T-Bill yield is precisely the
yield on a par floater of the US government.

With this understanding in place, let us now return to the only
possible explanation for the swap rate being less than the UST rate in
a perfect market – the asset swap floating leg must exceed Libor
(or the asset swap spread must be positive). In this case, in a
perfect market the fixed leg (which is the UST bond yield) must also
exceed the swap rate – the asset swap seller receives a larger
fixed leg than in an interest rate swap but also pays a higher
floating rate.

So the position is that for the current interest rates to be
consistent with a perfect market, the asset swap spread should be
positive while we know that the modified asset swap spread (the one
that terminates on default by the US government) is the negative of
the TED spread and is therefore expected to be negative. The
difference between the asset swap and the modified asset swap is that
after default by the US government, the modified swap terminates while
the ordinary asset swap subsists.

Everything now depends on what Libor is likely to be after the
default by the US government. If Libor is expected to be high, the
asset swap seller would have to make large floating rate payments in
return for the fixed rate payment from the asset swap buyer. The
subsisting swap would therefore be a liability to the asset swap
seller and he would therefore insist on paying a lower (more negative)
spread in the asset swap than in the modified asset swap where this
liability would not exist. This would imply that the asset swap
floating leg would be even lower than the T-Bill yield and therefore
much lower than Libor. The 30 year UST yield must therefore be less
than the swap rate.

For the 30 year US yield to be higher than the swap rate in a
perfect market therefore the asset swap must be beneficial to the
asset swap seller after the default by the US government. This can
happen only if interest rates are very low after default. I do not
find this very plausible. I would expect sovereigns to default on
local currency debt after inflation has been tried and found to be
wanting. With double digit inflation, one would imagine Libor also to
be in double digits and the asset swap would be a huge liability to
the asset swap seller who would be receiving something like 4.5%
fixed. Considering this liability, the asset swap spread should be
less than the T-bill yield which in turn is less than Libor.

Thus it appears to me that a 30 year swap rate less than the 30
year UST yield is consistent with perfect markets only if we are
willing to make either of the two assumptions:

1. The TED spread is expected to be negative implying that banks are
   safer than the US government; or
2. A potential default by the US government would happen in an
   environment of very low rates where Libor would be very low.

I find both these assumptions implausible and would believe that
the phenomenon that we are seeing in 30 year swaps is due to the
limits to arbitrage arising from inadequate capital and leverage.

One final question that might trouble the reader is the assumption
that there is no counterparty risk in the swaps even when the
sovereign itself has defaulted. Actually, if we simply assume that all
the swaps terminate on default by the US government, the above
arguments still go through. The fixed rate payer in the interest rate
swap makes money before the default. If at this point, he is allowed
to pack up his bags and go home, that is fine in this model.

This has been a difficult piece of analysis for me and I would
welcome comments, suggestions and corrections.