Building SOFR Forward Curves for Legacy LIBOR Swap Fallbacks

# Compounding arrear SOFR Forward rate/curve


# Compounding arrear SOFR Forward rate/curve












As per ISDA protocol and supplements, they stated that the fallback rate to be used on legacy derivative contracts is the compounding in arrears SOFR rate (based on a 2-day backshift) + a fixed spread adjustment (which will be published by Bloomberg). In other words, the derivate market will not actually make use of a 'forward-looking term SOFR' as with the case of the loan market (recommended by the ARRC).

My question is whether anyone knows how these rates will be used to value an interest rate swap. (e.g. we had a 3-month LIBOR swap, and now we are transitioning to a 90 day compounded in arrears SOFR + fix adjusted spread for 90 day period), usually, we would use a 3-month LIBOR Forward curve and discounting curve. But know if I am correct we would require a 90 day compounded in arrears SOFR Forward curve in order to value (not price) the swap.

I have not seen any announcements regarding the publication of this Forward curve (just forward looking rates, which is not the same thing as forward rates) or any methodology on valuing a swap using this fallback rate.

Also, If only seen SOFR swaps and OIS swaps, but NOT 30 day, 60 day or 90 day SOFR swap rate being published (unless I am missing something). So a forward curve will not be constructed based on the quoted swap rates.

## Answer by Jan Stuller (score 2, accepted)

https://quant.stackexchange.com/a/66587

Good question. There are two things to consider:

- SOFR Swap curve

- USD LIBOR Swaps that will fall back onto the 90-day backward-compounded spot SOFR + a fixed spread.

To my knowledge, SOFR swaps (i.e. 1 above) are already liquid and trade heavily: after all, London Clearing House switched to SOFR discounting from Fed-Funds discounting last year: these "standard" SOFR swaps work just the same way as the "old" Fed-Funds OIS swaps: the floating leg is compounded based on the daily published spot SOFR (compounded in arrears). But the entire swap curve then gives you an implied forward curve based on the fixed leg of these standard SOFR swaps (just the same way that a normal LIBOR swap curve gives you implied forward rates, see here).

Now as far as the exact mechanics of how the existing USD LIBOR swaps will fall back onto 90-day compounded SOFR + spread: somehow, the 90-day forward SOFR rates will have to be implied from the existing SOFR OIS swap curve: obviously, the granularity will be an issue here in the sense that the standard SOFR OIS curve to my knowledge trades with annual fixed coupons, so the granularity of the forward SOFR rates will be lower than the required 90 days.

However having said that, the quants building the curves have been at this for at least the past year and for sure they've figured out a way to do it: after all, all banks have by now mapped their existing LIBOR exposures onto fall-back curves, so the solution clearly exists (probably some kind of interpolation between the standard SOFR fixed coupons and / or convexity adjustment for the mismatch in frequencies).

So a specific procedure could be (Notation: $\lambda$ = annual fraction, for ease of notation, I write $\lambda$ instead of $\lambda_{(t_1,t_2)}$, obviously $\lambda$ will differ according to the annual fraction of the rate to which it's related, $DF(t_0,t_1)$ is the discount factor, and $r_{(t_0,t_1)}$ is the fixed SOFR swap rate quoted in the market, whilst $s_{(t_1,t_2)}$ is the implied SOFR forward rate between two points in time):

- Take the 3m SOFR swap quote (i.e. $r_{(t_0,3m)}$, could be 1y SOFR swap that was traded 9 months ago, now has 3 months maturity left): that's your first point. Add the fixed ISDA spread.

- Take the 6m SOFR swap quote (i.e. $r_{(t_0,6m)}$), solve for the forward SOFR $s_{(3m,3m)}$, i.e.: $DF_{(t_0,3m)}\lambda r_{(t_0,3m)}+DF_{(t_0,6m)}\lambda s_{(3m,3m)}=DF_{(t_0,6m)}\lambda r_{(t_0,6m)}$. Add the ISDA spread.

- Beyond 1-year, I assume you only have annual SOFR fixed quotes. Say you want to build $s_{(12m,15m)}$, $s_{(15m,18m)}$, $s_{(18m,21m)}$, $s_{(21m,24m)}$, but you only have quotes $r_{(t_0,12m)}$ and $r_{(t_0,24m)}$.

To make it simple, you could just assume that $s_{(12m,15m)}=s_{(15m,18m)}=s_{(18m,21m)}=s_{(21m,24m)}$ and solve:

$$DF_{(t_0,15m)}\lambda s_{(12m,15m)}+DF_{(t_0,18m)}\lambda s_{(15m,18m)}+DF_{(t_0,21m)}\lambda s_{(18m,21m)}+DF_{(t_0,24m)}\lambda s_{(21m,24m)}+DF_{(t_0,12m)}r_{(t_0,12m)}=DF_{(t_0,12m)}r_{(t_0,24m)}+DF_{(t_0,24m)}r_{(t_0,24m)}$$