# LIBOR Cessation: Construction of Term-RFRs as LIBOR Fallbacks; Forward vs. Backward Looking

This question emerged from comments in this feed: OIS rate to build Term structure.

I was wondering how the float leg of an IRS will look like in a post-LIBOR world. Assume the following time-line, where we have today ($$t$$) as the start of the accrual period and $$t+3m$$ as the payment date for the cashflow:

In a LIBOR world, the time-$$t$$ fixing is obviously known at $$t$$ so we know in-advance what amount we are due to pay/receive in 3 months from now. However, if we instead reference a risk-free rate (RFR) in our swap instead of the LIBOR, at $$t$$ we only know one fixing of the RFR. However, we need to transform it into a 3m term rate, and the question is how practicioners intend to do this.

It appears to me that there are two (three) main methods to compute the cashflow for this 3m accrual period:

1. Backward looking: Observe the daily RFRs from $$t$$ to $$t+3m$$ and then compound them in-arrears (for example observe the daily SOFR and compute the 3m rate at $$t+3m$$ as compounded realized daily SOFR, + spread). However, this means the exact cashflow due at time $$t+3m$$ is not known at $$t$$ but only at $$t+3m$$. In my opinion, the IRS would become what today is simply known as OIS.
2. Forward looking (scaled): This would take the time $$t$$ fixing of the RFR (for example SOFR published that morning, + spread) and "scale it" by the appropriate day count to make it a 3m rate.
3. Forward looking (compounded): Same as above, but at time $$t$$, use the compounded RFR between $$t-3m$$ and $$t$$ (for example compounded SOFR + spread).

NB: The last option is that a "forward looking" term rate for overnight rates can be derived from the respective futures market, once these are sufficiently liquid/reliable. But I'd like to ignore that case for the moment.

EDIT: This paper (chapter 3) provides a good overview in analytical form: https://ssrn.com/abstract=3308766.

• I've made an edit: please feel free to roll-back if you prefer the older version. Nov 17, 2020 at 8:13
• Good input, many thanks Jan. I have edited yet a little more, reflecting that option 3 is still "forward looking" in the sense that the $t+3m$ cashflow is already known at $t$. Nov 17, 2020 at 8:24

Another apology, I won't be able to give a definite answer either but in case of IR swaps I believe the following applies:

1. legacy (i.e. IBOR linked) contracts: the fallback protocol has been launched by ISDA last month and Bloomberg had been selected as fallback spread vendor a while back. In case LIBOR ceases to exist, the fallback rate is the compounded in-arrears risk free rate (I think a 2 day lookback applies as O/N rate fixings are published with a 1 day lag) plus a currency and tenor specific spread. More detailed information can be found here: https://www.isda.org/2020/10/23/isda-launches-ibor-fallbacks-supplement-and-protocol/ https://www.isda.org/2020/05/11/benchmark-reform-and-transition-from-libor/?_zs=siIqO1&_zl=lvcl5

2. new contracts: to my knowledge swaps referencing the new RFRs/ARRs at the moment are traded using general OIS conventions (again in arrears compounded floating rates but I think a payment lag of 2 days is used rather than a lookback). Of course some users will prefer fixing against a forward looking term rate but I have not seen any final concepts for those. I think there is a term SONIA under development and there are plans to get a forward looking SOFR term rate as well - but especially in the SOFR case liquidity in OIS might not be sufficient at the moment to come up with a IOSCO compliant way to set this term rate

• After some more information has been published and I gained further knowledge, I'll accept this answer because I think it correclty summarizes the two cases: namely, (a) legacy trades simply using ISDA fallback, which is a version of the observation-shifted in-arrears compound, and (b) new contracts, which from what I've seen are as close to plain vanilla OIS as possible - implying they opt for pay delays rather than lagged/shifted rate observation windows. Apr 8, 2021 at 10:52

Unfortunately, I cannot provide a definite answer.

In the major currencies, the risk free rate working groups (US:ARRC, UK:RFRWG and the EU:RFRWG) try to promote new standards for the cash and derivatives markets.

Further, there exist recommendations from various industry bodies how to incorporate (lagged) SONIA/SOFR(/ESTR) in new contracts. As an example, the Sterling WG is pressing for a Lookback without Observation Shift in the loan markets.

Although it is now a bit older, this paper by Marc Henrard summarised the issues quite nicely (from a quant perspective).

Sorry that I am not able to give a satisfying answer to your question, though... Maybe somebody else has some insight in which direction the latest deals (derivatives markets and cash markets) are headed. Also, I did not find ISDA's updated stance on this.

• You shouldn't be apologetic for such an informative answer! :) Nov 17, 2020 at 15:28
• Thank you! Adding to your point on the GBP working group; in Switzerland, the national WG proposed the in-arrears option (A) as well; however, they seem to go for the version that incorporates an observation shift. See here for a very nice summary by the Swiss National Bank: snb.ch/en/ifor/finmkt/fnmkt_benchm/id/finmkt_NWG_milestones. Nov 17, 2020 at 15:37

I believe that this recent paper by Andrei Lyashenko and Fabio Mercurio is going to help you! For me it was completely amazing. It seems that we can just extend the Libor Market Model in a "simple" manner to cope with the new RFR because we can define an extended numeraire $$P(t, T)$$ for $$t > T$$ that recovers Ibor-like properties, such as the martingale property, so you can analytically solve the floating leg fair value or present value.

I hope this helps! Thank you!

PD: I will just copy the Looking Forward to Backward-Looking Rates: A Modeling Framework for Term Rates Replacing LIBOR paper abstract:

In this paper, we define and model forward risk-free term rates, which appear in the payoff definition of derivatives, and possibly cash instruments, based on the new interest-rate benchmarks that will be replacing IBORs globally. We show that the classical interest rate modeling framework can be naturally extended to describe the evolution of both the forward-looking (IBOR-like) and backward-looking (setting-in-arrears) term rates using the same stochastic process. In particular, we show that the extension of the popular LIBOR Market Model (LMM) to the backward-looking rates completes the model by providing additional information about the rate dynamics not accessible in the LMM.

• Many thanks @rvignolo. I know this (excellent) paper very well in fact; and the notion of extended bond price is appealing indeed... Especially from a quant POV. However, my question is more about whether somebody (practicioners) knows how real-life contracts will be set-up (fwd or bwd looking), rather than how we can then price these contracts. In this step, the FMM is very useful of course. Nov 17, 2020 at 16:21
• Oh, I see! Even though it doesn't address exactly your question, it might be worth to keep this answer because someone could find it useful (if that is okay with you). Thanks! Nov 17, 2020 at 16:25
• Of course, and the suggested reference is worth a read for anybody who was interested enough to find this thread ;-) Nov 17, 2020 at 16:27