# Difference between 1-month and 3-month SOFR, and the relationship between them

I want to better understand the difference between the 1-month and 3-month SOFR rate, and when to use each for cash flow discounting.

For example - I have a 5 year project, and someone has decided to use the 3-month SOFR rate. My question is why? What does this assume about the movement of cash flows?

Furthermore, I would like to know if there's a mathematical relationship between the two (1-month vs 3-month).

Any useful web links are appreciated.

Thank you very much

EDIT: Some Googling led to the following definition

Three-Month Term SOFR means the rate for Term SOFR for a tenor of three months that is published by the Term SOFR Administrator at the Reference Time for any interest period, as determined by the Calculation Agent after giving effect to the Three-Month Term SOFR Conventions.

Does this definition provide any leads? Can I define the 3-month SOFR as $$F(t,T)$$ to be the forward rate of the 3-month SOFR at some forward time $$t$$?, and $$T=3$$ is fixed?

Are the forward curves then shown in the columns below? First, there is an FAQ for CME Term SOFR here. The input data for Term SOFR is 13 consecutive 1M SOFR futures and 5 consecutive 3M SOFR futures (both are enough to cover a year, the max tenor in the CME Term SOFR).

Based on the definition, I think it is the (annualized) expected future daily SOFR over the respective tenor. Since it is calculated from futures prices, the expectation should be under the risk-neutral measure. Let $$r_t$$ be the overnight SOFR at time $$t$$. Then, for a tenor of $$\tau$$ days, the CME Term SOFR may be defined as $$R_{t,t+\tau}=\mathbb{E}^Q\left[\prod_{s=t}^{s=t+\tau-1}\left(1+\frac{d_sr_s}{360}\right)\right] - 1$$ where $$d_s$$ takes care of the day count (more here). The product is the compounded daily SOFR rate. $$\tau$$ could be, e.g., 1 month or 3 months.

Roughly speaking, you can consider the term SOFR as the "true" risk-free interest rate. Particularly to your example, to calculate the NPV of a 5-year project, I don't think either 1M or 3M term SOFR is a good discount rate:

• The maturity should better be matched, so you should at least use 5-year zero yield calculated from SOFR swaps and futures;
• The project's cash flows are likely to be stochastic, so a risk premium on top of SOFR should be considered.

My thought on their methodology: On each day (end of Dec22, e.g.), the input is prices of consecutive 13 1M SOFR futures and 5 3M SOFR futures. Of course there is overlap between their coverage, e.g. some three 1M SOFR futures may cover similar time period as some 3M SOFR futures. Then, there are several methods to obtain implied zero yield curve from futures: bootstrapping, cubic spline, or Nelson-Siegel-Svensson model, using least squares or maximum likelihood estimation. The output will be the zero yield for every maturity, i.e. a mapping from maturity to SOFR zero yield.

• Hi @L. Francis Cong. – thank you very much for your reply. Your knowledge is excellent, and I appreciate your time. Is it possible that you are missing a $-1$ at the end, thereby giving $R_{t,t+\tau}$ a rate intepretation? That is $$R_{t,t+\tau}=\mathbb{E}^Q\left[\prod_{s=t}^{s=t+\tau-1}\left(1+\frac{d_sr_s}{360}\right)\right]-1$$ With this definition I believe I understand what $X$-M SOFR represents. $X=\tau$ in your explanation, and it is simply the forward interest rate, starting at $t$ and maturing at $\tau$ periods later. Ah - yes - we add a risk-premium to the SOFR. Jan 27 at 13:44
• @GustavoLouisG.Montańo Yeah. You are right. I've modified my answer. Jan 27 at 14:31
• Fantatsic. One last question: While $\tau$ is fixed in our example, in reality, there are a set of dates that $\tau$ needs to be applied to. How is this done? In other words, perhaps, say I have SOFR term structure. How would I derive the 3M curve from this? I'll be sure to read the documentation available, though, curious to hear your thoughts. Jan 27 at 14:50
• @GustavoLouisG.Montańo I am not sure if I understand your question, but I put my thought on their method in the main answer. Also I don't think their detailed methodology is publically available given that it is proprietary. Jan 27 at 20:49

The building blocks of these 1 month and 3 month rates are expected overnight rates. Overnight SOFR rate has a direct relationship to fed target rate. SOFR o/n currently @4.31 (-19 bps to the 4.5 target). Both the 1month and 3month SOFR rates are defined as the expected value of the average daily SOFR rate, so they could be different if the FOMC is expected to change the target rate within this time frame (between the end of 1st month and end of 3rd). Maybe in this project, the analyst evaluated that the 3 month rate is god proxy for the 5 month discount rate although the most agnostic thing to do would me to use the market price for the 5 month rate.