Get a detailed look at the Fed Funds futures probability tree calculations used by the FedWatch Tool to determine the probabilities of a Fed rate move at upcoming FOMC meetings.

## Assumptions:

- The probability of a rate hike (or conversely, a rate cut) is calculated by adding the probabilities of all target rate levels above (or below) the current target rate.
- Probabilities of possible Fed Funds target rates are based on Fed Funds futures contract prices, assuming that rate hikes/cuts are uniformly sized in increments of 25bps (0.25%), and that the Effective Fed Funds Rate (EFFR) will react proportionally to the size of the hike/cut.
- FOMC meetings occur on a published schedule with a mostly even distribution, of which one month per quarter does not contain a meeting.
- For months with meetings, the corresponding monthly Fed Funds futures contract price is used to calculate the probabilities of hikes/cuts.
- The projected end rate for EFFR in each month should be equal to the start rate for EFFR in the subsequent month.
- Calculated probabilities are estimates based on these assumptions and may be subject to variation if any tenet is compromised.

## Overview:

The FedWatch Tool calculates *unconditional probabilities* of Federal Open Market Committee (FOMC) meeting outcomes to generate a binary probability tree. CME Group lists 30-Day Federal Funds (ZQ) futures, prices of which incorporate market expectations of the average daily Effective Federal Funds Rates (EFFR) during the futures contract months (e.g., the market price of ZQU2 reflects the consensus expectations of the average EFFR during the month of September 2022). The EFFR is published by the Federal Reserve Bank of New York each day. The EFFR represents a transaction-volume weighted average of the previous day’s rates on trades arranged by major brokers in the market for overnight unsecured loans between depository institutions.

In the FedWatch Tool’s probability analysis, the calculations assume that the size of a rate change is always in multiples of 25 basis points (bps), and that EFFR is bounded below by zero. Because the price of each Fed Funds futures contract represents the expected average daily EFFR for that contract month, if one were in a FOMC meeting month where there was no meeting in the prior month, then the futures price of the previous month is independent of the outcome of current month’s meeting.

Likewise, if one were in a FOMC meeting month such that there was no meeting scheduled for the following month, then the futures price of the following month contains only information about the outcome of the current month’s meeting. If one assumes that in its current month meeting the FOMC will decide either to raise its daily EFFR target or to maintain the status quo, then the probabilities of a rate hike versus no rate hike would be calculated as:

P(Hike) = [ EFFR(end of month) – EFFR(start of month ) ] / 25 basis points

P(NoHike) = 1 – P(Hike)

## Methodology:

To calculate the unconditional probability of a change in the target rate at the current month’s FOMC meeting, one must first start with the nearest full month *without* a FOMC meeting.^{1} For a given month without a FOMC meeting, that month’s futures contract price represents the average EFFR rate for the entire month. Thus, for a month T without a FOMC meeting, one can assume that **EFFR(End) _{T-1} = EFFR(Avg)_{T} = EFFR(Start)_{T+1}**. The table below illustrates some of these relationships:

Source: CME Group

Using these formulas, *and starting from a month without a FOMC meeting*, one can progressively build up probabilities for the outcomes of each upcoming FOMC meeting. The probability is calculated as follows:

- First, start by looking at the nearest month
*without*a FOMC meeting. Given the EFFR(End)_{T-1}= EFFR(Avg)_{T}= EFFR(Start)_{T+1}relationship described above, fill out the EFFR(End) and EFFR(Start) for the months immediately preceding and following the non-FOMC meeting month. - Then, for those months immediately before and after a non-FOMC meeting month, use the formulas in the table above to calculate the EFFR(Start) or EFFR(End) by using the EFFR(Avg) and the EFFR(Start or End) which were copied from the non-FOMC meeting month.
^{2} - Follow step 2 above successively until every month has values for EFFR(Start), EFFR(End), and EFFR(Avg), remembering the EFFR(Avg) is simply 100 – current contract price for said month.
- Then calculate the expected monthly change in EFFR for the nearest upcoming month with a FOMC meeting: EFFR(End) – EFFR(Start).
- Then, calculate the equivalent number of 25bp hikes or cuts: (Expected EFFR Change)/0.25.
- Break down the number of expected 25bp hikes or cuts into its whole integers and remaining decimals (e.g., 2.1103 expected 25bp hikes 2 hikes + 0.1103 hikes). In mathematical terms, these are also called the “characteristic” and the “mantissa,” respectively.
- The whole integer obtained above determines the lower bound of potential rate hikes or cuts. That is, for the number used above, the minimum size of a rate hike expected by the market is 2 x 25bps = 50bps. The probability of a hike of this size can be calculated as 1 – remaining decimals (e.g., 2 hikes + 0.1103 hikes Prob(50bps hike) = 1 – 0.1103 = 0.8897 = 88.97%).
- The probability of a rate hike of a larger size than that of the integer we calculated above is simply equal to the remaining decimals. (i.e., given an expected number of 25bp hikes of 2.1103, the probability of a hike larger than 50bps is 0.1103, or about 11%).

The eight steps above demonstrate how to calculate the probabilities on the first node of this binary tree – the nearest upcoming FOMC meeting. As a result, these calculations will produce probabilities for two distinct outcomes: the probability of the expected hike size, and the probability of a hike size 25bps larger than the expected. The inverse outcomes can be found if the market expects a rate cut: the probability of the expected cut size, and the probability of a cut size 25bps larger than expected. Additionally, for expected EFFR changes of less than 25bps, this tool calculates the probability of a hike/cut of 25bps, and the probability of no hikes/cuts.

The following FOMC meeting month will represent the second node of this probability tree. Following the steps described above, one can calculate the binary probabilities of the expected hike size and a larger hike than expected. Finally, having calculated probabilities for two successive FOMC meetings, one can combine these probabilities into cumulative probabilities. By multiplying the probabilities of each outcome on the first upcoming FOMC meeting with those of the following meeting, one can arrive at three or four distinct possible outcomes for the second meeting, as well as the cumulative probabilities of each outcome.

For example, if the market expects a 75bp or a 100bp hike on the next FOMC meeting, and a 50bp or a 75bp hike on the following meeting, the three distinct cumulative outcomes that one can calculate are: a total hike of 125bps; a total hike of 150bps; and a total hike of 175bps. With each successive FOMC meeting added to this probability tree, the probability of more outcomes can be calculated.

__Example__: September 21, 2022 FOMC Meeting

To calculate the probabilities for this meeting, first look at the nearest upcoming month without a planned FOMC meeting: October 2022:

ZQV2 Price = 96.9400

ZQV2 Implied Avg. EFFR Rate = 100 – 96.9400 = 3.0600

For months __without__ FOMC meetings, EFFR(End)_{T-1} = EFFR(Avg)_{T} = EFFR(Start)_{T+1}

Thus, the EFFR(End)_{Sept} = EFFR(Avg)_{Oct} = 3.0600

For months *with* FOMC meetings, and *no* FOMC meetings in the subsequent month, EFFR(Start)_{T} = { EFFR(Avg)_{T} – [ (M/M+N) * EFFR(End)_{T} ] } / (N/M+N)

ZQU2 Price = 97.4475

ZQU2 Implied Avg. Rate = 100 – 97.4475 = 2.5525

Days before FOMC Meeting (N) = 20

Days after FOMC Meeting (M) = 10

EFFR(Start)_{Sept} = { 2.5525 – [ (10/30) * 3.0600 ] } / (20/30)

EFFR(Start)_{Sept} = 2.2987

Having calculated EFFR(Start)_{Sept} and EFFR(End)_{Sept}, one can now calculate the expected EFFR change during the month of September:

EFFR(End)_{Sept} – EFFR(Start)_{Sept} = 3.0600 – 2.2987 = 0.7613

In multiples of 25bps, 0.7613 represents 3.0450 hikes of 25bps size. Broken down into whole integers and remaining decimals, one arrives at three full hikes and 0.0450 of a hike.

The two outcomes, then, which one can calculate probabilities for are 1) a hike with the size of 3 * 25bps = 75bps, and 2) a hike of 75bps + 25bps = 100bps:

Prob(75bp hike) = 1 – Remaining decimals = 1 – 0.0450 = 0.955 = 95.5%

Prob(100bp hike) = Remaining decimals = 0.0450 = 4.5%

Following the same procedure for November, using the implied avg. rate from October to calculate the EFFR(Start)_{Nov}, one can calculate the following *unconditional* probabilities:

Prob(50bp hike) = 1 – Remainder = 1 – 0.1103 = 0.890 = 89.0%

Prob(75bps hike) = Remainder = 0.1103 = 11.0%

Finally, one can combine the probabilities from the September meeting with those of the November meeting to arrive at *conditional/cumulative* probabilities of certain outcomes:

Prob(cumulative 125bp hike) = Prob(75bps)_{Sept} * Prob(50bps)_{Nov}

= 0.955 * 0.890 = 0.850 = 85%

Prob(cumulative 150bp hike) =

(Prob(75bps)_{Sept} * Prob(75bps)_{Nov}) + (Prob(100bps)_{Sept} * Prob(50bps)_{Nov})

= (0.955 * 0.110) + (0.045 * 0.890) = 0.105 + 0.040 = 14.5%

Prob(cumulative 175bp hike) = Prob(100bps)_{Sept} * Prob(75bps)_{Nov}

= 0.045 * 0.110 = 0.005 = 0.5%

This process can be repeated successively for all further FOMC meeting months for which CME has listed Fed Fund futures contracts listed.

## References

- If the current month does not contain an FOMC meeting, it does not become the first anchor month. In such circumstance, the next
*full*month that does not contain an FOMC meeting is designated as the first anchor for calculations. - The FedWatch Tool follows a specific order of operations in relation to these
*anchor*non-FOMC months. Given that the contract prices for each month are independent from each other, for every non-FOMC month there exists the possibility of a small discontinuity in the path of implied rates. The approach that the tool uses to minimize these discontinuities is to specify the*directions*in which the implied rates from non-FOMC months are propagated. For every non-FOMC month T, the FedWatch Tool utilizes the EFFR(Avg)_{T}to populate the EFFR(End)_{T-1 }and the EFFR(Start)_{T+1}. Crucially, considering these adjacent months, the tool continues its calculations in reverse-chronological order – that is, the calculated EFFR(Start)_{T-1 }is copied to populate the EFFR(End)_{T-2}, but the calculated EFFR(End)_{T+1}is not used to populate the EFFR(Start)_{T+2}. Alternatively stated, implied rates from non-FOMC anchor months propagate forward in time for only one month and backwards for as many months needed until another non-FOMC anchor month is reached.

*All examples in this report are hypothetical interpretations of situations and are used for explanation purposes only. The views in this report reflect solely those of the author and not necessarily those of CME Group or its affiliated institutions. This report and the information herein should not be considered investment advice or the results of actual market experience. *