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.

Posted at 4:30 pm IST on Wed, 16 Sep 2009         permanent link

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