Post-LIBOR Swap Cash Flows Using Risk-Free Rates

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


# 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:



- 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.

- 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.

## Answer by TCUK2020 (score 2, accepted)

https://quant.stackexchange.com/a/59462

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

- 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

- 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

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/59437

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.

## Answer by rvignolo (score 3)

https://quant.stackexchange.com/a/59439

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.