Negative Swap / Government Spread: A SOFR definition impact

Negative government bond spreads have been a puzzle for some people for a long time. The question has been stated in some places as ``How can the banks borrow at a lower rate than the government?''.

I have already partially given my point of view about that issue in a blog 10 years ago. That was in the context of the LIBOR swaps, but it is still true in the SOFR case.

Now I want to add some technical details for the SOFR OIS case. To some extent, I claim that the swap spread have to be negative!

SOFR-OIS swap / government spreads

In this analysis, I'm using the results of the multi-curve framework in a loose sense. I'm using expected values, without clarifying in which measure they are and extended the results to government bonds. The goal is to indicate that the ISDA definition of SOFR (introduced below) used in swaps may have an hidden impact on swap spreads.

For this simplified approach, I'm presenting the impact only for zero-coupon bonds. The notation are

  • ri for daily fixing of overnight rate according to ISDA defined SOFR,
  • R for the term OIS rate of maturity T,
  • gi for the overnight government rate,
  • G for the term government rate

The swap rate satisfies

and the government rate satisfies

So to compare R and G, it ``suffices'' to compare ri and gi.

Let's start by the normal situation where there is no government issue and the repo market is functioning normally. In that case we can consider that ri ≅ gi. The private lending behind the (reverse)-repo is guaranteed by a government security. Its rate should slightly below government rate as there is a double guarantee: private and government. On the other side, in case of default of the private borrower, some extra work is required to recover the cash. Moreover, there is no exact overnight government rate as there is no overnight security market. But all in all, up to a couple of basis point, we can suppose that in that case, ri ≅ gi.

Now we move the ``not so normal situation'' where there is a technical default of the government. By technical default, I mean a situation where the government is able to pay but not willing to do so for some technical or political reason. The current situation in the U.S. -- government shutdown since 1 October -- is exactly the type of mechanism that could lead to a technical default. The short rate on government paper could sky rocket. One is never sure to be paid tomorrow, even if one is certain to be paid the day after, the annualised rate requested for such a lending will be a lot higher. So we have gi higher for a certain period of time. On the other side, ri is a repo rate where the first line is the private company, which is still obliged to pay, even if there is a government problem. What could also happen is that the repo market does not work anymore for a short time. In that case the repo benchmark does not exist anymore and the Federal Reserve Bank of New York does not publish SOFR (and OBFR) for some time. In that case, according to the ISDA definition of SOFR — Supplement number 57 to the 2006 ISDA Definitions —, the OIS compounded rate will be based on the FOMC Target Rate. There is no reason for that rate to be as high as the government rate as it is targeting lending rate between prime quality banks. Putting those possible situations together, it is not impossible to have

gi ≫ ri
for some period of time.

Take a simple example where all overnight rates are 5.00% for 5 years. In case of trouble, the government rate goes up to 10.00%. A 6-month trouble generate a government rate over the 5-year period equivalent to 5.50%.

Obviously to make it a real quantitative model, by opposition to the small toy above, one would need to quantify the probability of the trouble case, the overnight rate in case of trouble, and so on.

One thing that this note does, is to show that hidden in the definitions of financial products, some features may have large financial impacts. User beware! (And get advise by subject matter experts)

This text is also available as a muRisQ Market Infrastructure Analysis.

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