In the post-LIBOR world, forward-looking SOFR rates will be needed to help price financial instruments. In addition, forward-looking term SOFR rates are part of the Alternate Reference Rate Committee's (ARRC) Paced Transition Plan. In the note, we describe how the SOFR rate and SOFR futures can be used to construct 1) a daily SOFR forward curve with jumps on key dates such as FOMC meetings and 2) term SOFR forward curves with few to no jumps. Curves constructed from this are available at Chatham Rates.
The Federal Reserve Bank of New York started publishing the new benchmark Secured Overnight Financing Rate (SOFR) interest rate in April 2018, and future contracts started trading in May 2018. With this data we can bootstrap the daily SOFR forward curve that represents the market-expected SOFR fixings in the future. Unlike Chatham’s other bootstrappers, the jump bootstrapper allows for jumps on key dates such as FOMC meetings. These jumps reflect the high correlation between SOFR and the Fed target rate. Also these jumps reflect the transaction-based instead of quote-based nature of the SOFR rate. From this daily SOFR forward curve, we then build the forward curves for term rates including the monthly and quarterly forward curves which are generally smoother than the daily SOFR curve.
Here we describe the jump bootstrapper that uses the market data to build a forward curve with jumps on key dates.
To build the SOFR forward curve we use the following market data: historic SOFR fixing rates, SOFR future rates, and Federal Open Market Committee meeting dates.
We use SOFR fixing rates, SOFR 1-month futures contract rates and SOFR 3-month futures contract rates.
The Fed has eight scheduled meetings each year, or about once every six to seven weeks, to determine the Fed Funds Target Rate. Most rate changes occur during the meetings in March, June, September, and December. Additional meetings are scheduled as necessary.
For SOFR, it was useful to incorporate new features into the bootstrapping process.
1. The most important new idea is that key dates play an important role in the SOFR rate. In analyzing historic indicative SOFR rates, it was observed that the SOFR rate jumped after the Fed changed rates. These jumps were approximately equal to the change in the Fed Funds target rate.
2. The second idea is that the SOFR rate does not on average seem to rise or fall ahead of Fed rate changes. Between meetings, the SOFR rate changes from day to day due to market volatility, and the rate often jumps at month end due to bank regulatory requirements. The SOFR rate, however, does not significantly drift up or down in value between FOMC meetings.
Our model combines these two observations and builds the daily SOFR forward curve. The key dates are built as follows. For the first seven months, we find the start of the first 1-month futures contract and any FOMC meeting dates in between the start and end of the 1-month futures dates. For the remaining five years, we find any FOMC meeting dates, as well as the beginning and end of the 3-month contracts. In between these key dates, the SOFR rate is free to take any constant value.
The SOFR rates between the key dates are then chosen to be a constant value that best matches the 1) the 1-month SOFR futures prices in the front end of the curve and 2) the 3-month SOFR futures prices in the rest of the curve.
Model Approximate Solutions
The model described has more futures prices to match than there are key date intervals. For example, there are more 1-month futures prices than Fed meetings in the nearest six months; the Fed meets approximately every 6 weeks. Therefore, this model generally does not exactly reproduce all the futures contract prices.
It is possible, however, to find a curve that reproduces all the futures contract prices within an acceptable error. An example of the curve is shown below in Fig. 1:
In our analysis the differences between the SOFR forward curve and the futures prices were typically small when the futures contracts had sufficient trade volume. When the trade volume was low, then the differences could be larger but still usually smaller than the bid-ask spread for the contract. Overall we think these differences in SOFR forward curve and futures are in line with market liquidity of the futures contracts.
Like most forward curves that rely on futures data, we also incorporate a convexity adjustment to account for the impact of futures daily settlement. The convexity adjustment can be derived with different underlying assumptions and in our model we used the Flavell convexity adjustment with a five-year flat vol.
The SOFR daily rate is not a direct replacement for LIBOR which has interest rates ranging from one day to one year. For many applications that need a term rate, a historic term rate that is built from historic fixing rates will be sufficient. However, from the daily SOFR forward curve, we can create forward-looking term rates of different tenors.
So far we have created a monthly SOFR forward curve that is a simple average of the daily SOFR forward values. We have also created a quarterly SOFR forward curve that is compounded from the daily SOFR forward values. Both of these term rates are forward looking, are generally smoother than the daily SOFR forward curve, and will start rising or falling ahead of the jumps in the daily curve. Other term rates can be easily generated with the daily SOFR forward curve.
The availability/liquidity of term SOFR rates is a critical issue for end users planning for transition to LIBOR alternatives. Regulators and the Alternative Reference Rates Committee (ARRC) appear to have expressed willingness to support term SOFR rates for use in certain types of cash products, but have been less clear about whether term SOFR rates will be supported for use in derivatives used to hedge cash instruments referencing term SOFR.
The SOFR landscape has evolved sufficiently that it is possible to consider how a term SOFR rate might be constructed and gain sufficient liquidity to be viable. Likewise it is possible to begin assessing the alignment of potential term SOFR rates with IOSCO principles and the susceptibility of such rates to regulatory concerns such as manipulation.
A term SOFR rate that is derived from Designated Contract Market (“DCM”) or Swap Execution Facility (“SEF”) market transactions, for example, offers safeguards against manipulation. Provided the underlying markets had sufficient liquidity, such a rate might be competitively priced when compared to potential compounded SOFR rates. On the other hand, if there are wide trading spreads for derivatives tied to term SOFR, it may be that such costs would outweigh the operational advantages of term SOFR rates.
Recap and Future Directions
The jump bootstrapped curve reflects new features of SOFR. The SOFR forward curve:
- Is constant between key dates such as FOMC meetings
- Represents the average SOFR rate between key dates
- Is free to have arbitrary-sized jumps on the key dates
- Currently is built from 1-month and 3-month SOFR future prices
- Ties out to the futures prices quite reliably
- Can be used to build different term rates
We can imagine making changes to this model in the future. If necessary, for example if the model differences start to get larger, we can better match the futures prices. One possibility is to add additional key dates so the curve can more flexibly fit the market data; one could explore adding key dates at month end to account for the SOFR spikes at month end. Also we currently build the short end of the curve using only 1-month futures. In the future we can use overlapping 1-month and 3-month futures, instead of treating those parts of the curve separately. Finally, when SOFR swaps become liquid, they could be incorporated into the curve construction.
There are a number of outstanding questions with the new model. While this model seems to capture new and important features of SOFR, it is possible that many other market participants will build a SOFR curve without jumps. In the future, as the market develops a consensus on how to build the SOFR forward curve, we will revisit the SOFR forward curve and see if we need to update this model.
 In comparison, LIBOR anticipates interest rate changes and rises or falls ahead of an anticipated rate change by the Fed. Therefore, a smooth forward curve is appropriate for LIBOR.
 Building a curve that incorporates overlapping futures could also be done as discussed in the recap.
 In the recap, we discuss how increasing the number of key dates would allow us to exactly recover the futures contract prices.