How the FedWatch Tool calculates probability trees for multiple Fed meetings
Mathematical foundations of the CME Group's expanding binary probability tree framework
The CME FedWatch Tool uses an "expanding tree" structure to calculate probabilities of Federal Reserve rate decisions. The method is called "expanding" because it builds a branching structure that grows with each successive FOMC meeting, mapping out all possible sequences of rate changes.
Each FOMC meeting presents two primary outcomes: either the Fed changes rates by 25 basis points (up or down), or rates remain unchanged. After one meeting, there are two possible rate levels. After two meetings, there are three possible rate levels (but four different paths to reach them). After three meetings, there are four possible rate levels, reached through eight different paths.
This combinatorial growth—where each meeting doubles the number of paths—creates the "tree" structure. The CME methodology assigns probabilities to each branch based on Federal Funds futures prices, then traces all possible paths forward to calculate the likelihood of different rate outcomes several meetings ahead.
The CME method calculates the probability of each path through this tree using futures prices. It's called the "gold standard" because it's transparent, systematic, and used worldwide.
The CME FedWatch Tool employs an expanding binary probability tree to extract market-implied probabilities of FOMC rate decisions from 30-Day Federal Funds futures prices. This methodology represents the most widely-referenced derivative-based approach to monetary policy expectations extraction.
Core Innovation: The expanding tree framework elegantly addresses the challenge of converting continuous futures price information into discrete probability distributions over multiple sequential policy decisions. By imposing structure (binary branching at each node) while maintaining flexibility (adaptive to market pricing), the methodology balances tractability with market responsiveness.
Theoretical Foundation: The approach rests on the fundamental theorem of asset pricing, which establishes the existence of a risk-neutral probability measure under which futures prices equal expected spot rates. For Fed Funds futures with deterministic short rates over the contract period, this simplifies to:
This page provides comprehensive technical documentation of the CME expanding tree methodology:
For the CME method to work, it needs to make some simplifying assumptions. These aren't always perfectly true, but they're close enough most of the time to produce good predictions.
What it means: The Fed moves rates in quarter-point (0.25%) increments
Reality check: Usually true! The Fed loves 25bp moves. But in emergencies (like 2022), they sometimes do 50bp or 75bp moves.
What it means: When the Fed raises its target by 25bp, the effective fed funds rate (what actually trades in the market) also rises by 25bp
Reality check: Very close to true under the current ample reserves system
What it means: Interest rates can't go below zero
Reality check: True for the US. (Some other countries like the ECB have had negative rates, but that's a different story.)
What it means: At each Fed meeting, only two things can happen - either what the market expects, or one step different (up or down 25bp)
Reality check: This is a simplification. Sometimes the market is genuinely uncertain between three outcomes.
What it means: The Fed only changes rates at their scheduled 8 meetings per year, never between meetings
Reality check: Usually true. Emergency inter-meeting moves are rare (last one was March 2020 during COVID)
What it means: The rate at the end of one month equals the rate at the start of the next month
Reality check: True! Rates don't jump overnight between months.
What it means: Futures prices reflect what traders expect to happen, not what they fear or hope for
Reality check: Not quite! Research shows futures prices include a "risk premium" - traders pay a bit extra for insurance. We'll discuss this later.
The CME expanding tree methodology rests on seven fundamental assumptions that constrain the probability extraction problem to a tractable form. Understanding these assumptions is critical for assessing when the methodology provides reliable guidance and when alternative approaches become necessary.
Justification: The Federal Reserve has demonstrated a strong preference for quarter-point moves since the mid-1990s, reflecting a desire for gradualism and predictability in policy implementation.
Violations: The assumption breaks down during crisis periods when the Fed executes larger moves (50bp or 75bp changes occurred in 2001-2002, 2008, and 2022-2023). The methodology adapts by computing probabilities for larger increments, but the binary tree structure cannot represent genuinely trimodal distributions where significant probability mass exists on three distinct outcomes.
Justification: Under the current ample reserves framework with Interest on Reserve Balances (IORB) as the primary tool, EFFR tracks IORB (the midpoint of the FOMC target range) with minimal spread, typically 1-5 basis points.
Historical Context: This assumption is regime-dependent. It holds well under ample reserves (2020-present) but would not have held during the pre-2008 corridor system or during the scarce reserves regime of 2017-2019.
Justification: In the U.S. institutional context, negative nominal interest rates face legal and operational obstacles. The Federal Reserve has consistently stated that negative rates are not considered a viable policy tool.
Cross-Country Note: This assumption does not hold universally - the ECB, Bank of Japan, Swiss National Bank, and others have implemented negative policy rates. Applications of CME-style methodologies to these jurisdictions require modification.
Justification: The binary structure simplifies computation dramatically. At each node, the market can assign probability \(p\) to one outcome and \((1-p)\) to another, extractable from the fractional part of the expected rate change.
Limitations: This is the methodology's most significant oversimplification. During periods of genuine uncertainty (e.g., early 2023 when markets debated between hold/hike/cut), restricting to two outcomes distorts the probability distribution. The tool cannot natively represent scenarios where \(P(\text{outcome } A) = 0.4\), \(P(\text{outcome } B) = 0.35\), and \(P(\text{outcome } C) = 0.25\).
Justification: Inter-meeting moves are historically rare, occurring only during extreme circumstances (9/11, 2008 financial crisis, March 2020 COVID crisis). Their rarity justifies excluding them from baseline probability calculations.
Failure Mode: During acute crises when inter-meeting action becomes possible, futures markets may price in probabilities that the methodology cannot properly decompose, leading to inconsistent probability estimates.
Justification: Rates do not jump discontinuously at month transitions. This continuity condition allows the methodology to propagate rate information forward and backward across non-FOMC "anchor" months.
Technical Role: This assumption is critical for the algorithm's propagation rules and provides the constraint equations needed to solve for starting and ending rates within FOMC months.
Justification: Standard derivatives pricing theory establishes that futures prices reflect risk-neutral expectations. This assumption allows direct extraction of probabilities from price levels.
Critical Caveat: Extensive empirical literature (Piazzesi & Swanson 2008; Hamilton & Okimoto 2011) documents that Fed Funds futures contain significant positive risk premia averaging 35-61 basis points per year, which are countercyclical and predictable. The methodology extracts risk-neutral probabilities, not physical probabilities. For policy forecasting (as opposed to measuring market perceptions), risk premium adjustment becomes essential.
These seven assumptions collectively define the CME methodology's domain of applicability:
Now let's walk through exactly how the CME method calculates probabilities. We'll break it down into simple steps.
We want to know: What's the probability the Fed will raise, lower, or hold rates at their next meeting?
To figure this out, we use:
An anchor month is a month with NO Fed meeting. These are super helpful because they're simple - the rate doesn't change all month! The futures price directly tells us what the rate will be.
Example: If October has no Fed meeting and the October futures price is 96.94, then we know the average rate for October will be 100 - 96.94 = 3.06%.
Look at the Fed's meeting schedule. Find months without meetings. These give us fixed points.
Example: If the Fed meets in September, November, and December, then October is an anchor month.
For months with Fed meetings, figure out what the rate is at the start of the month (before the meeting).
We use the anchor month to help us. Since the rate at the end of September equals the rate at the start of October (that's the continuity assumption), we can work backwards.
The futures price tells us the average rate for the whole month. Since we know the starting rate and how many days are before vs. after the meeting, we can calculate what the ending rate must be.
Formula: Ending Rate = (Average Rate × Days in Month - Starting Rate × Days Before Meeting) ÷ Days After Meeting
Simple subtraction: Expected Change = Ending Rate - Starting Rate
This tells us how much the market expects the Fed to move rates.
Divide the expected change by 0.25 (since the Fed moves in 25bp increments).
Example: If the expected change is 0.725%, then 0.725 ÷ 0.25 = 2.9
Split that number into two parts:
Then:
In this case: 10% chance of 50bp hike, 90% chance of 75bp hike
Repeat the whole process for the next Fed meeting, using the ending rate from this meeting as your new starting point.
The CME methodology proceeds through seven systematic steps to extract probabilities from futures prices. Let us formalize each step mathematically.
Define the set of FOMC meeting dates:
A month \(t\) is an anchor month if:
For anchor months, the relationship is direct:
The continuity assumption establishes:
This provides boundary conditions for solving the system. If month \(t\) is an anchor with \(t+1\) containing an FOMC meeting:
For month \(t\) containing an FOMC meeting on day \(d\) (with \(n\) total days), the futures settlement rate represents the volume-weighted average:
Solving for the post-meeting rate:
Express \(x_t\) as sum of integer and fractional parts:
Under the binary branching assumption, the risk-neutral probabilities are:
For meeting \(i+1\) following meeting \(i\), recursively apply the procedure using:
Cumulative path probabilities multiply along branches:
The methodology employs asymmetric propagation to minimize discontinuities:
This design reflects that backward propagation uses realized constraints while forward propagation would amplify forecast uncertainty.
Let's work through a real example to see exactly how this works. We'll use the September 21, 2022 Fed meeting - a fascinating case because the Fed was hiking rates aggressively to fight inflation.
Futures Prices:
October has no Fed meeting, so it's simple:
Average rate for October = 100 - 96.9400 = 3.0600%
This rate stays the same all month, so:
September has 30 days. The Fed meeting is on September 21.
September futures price tells us the average: 100 - 97.4475 = 2.5525%
Now we solve for the starting rate. We know:
Formula: Average = (Days Before × Start Rate + Days After × End Rate) ÷ Total Days
Rearranging:
Start Rate = (Average × Total Days - Days After × End Rate) ÷ Days Before
Start Rate = (2.5525 × 30 - 10 × 3.0600) ÷ 20
Start Rate = (76.575 - 30.600) ÷ 20
Start Rate = 45.975 ÷ 20 = 2.2988%
(Note: The CME got 2.3350% using slightly different day counts. The principle is the same!)
Expected Change = End Rate - Start Rate
Expected Change = 3.0600 - 2.3350 = 0.7250% or 72.5 basis points
72.5 ÷ 25 = 2.9
Split this into:
Probability of (2 × 25bp = 50bp hike) = 1 - 0.9 = 0.10 or 10%
Probability of (3 × 25bp = 75bp hike) = 0.9 = 0.90 or 90%
Market-implied probabilities for September 21, 2022 FOMC meeting:
What actually happened: The Fed hiked by 75 basis points! The market got it right.
This example demonstrates the CME methodology using actual market data from September 2022, during the Federal Reserve's aggressive inflation-fighting hiking cycle.
Date of Analysis: September 21, 2022
FOMC Meeting Schedule:
Futures Contract Prices:
Phase 1: Establish Anchor Constraints
October 2022 contains no FOMC meeting, establishing it as an anchor month:
By continuity:
Phase 2: September Within-Month Decomposition
Meeting parameters:
Implied average rate:
Solve for starting rate using the within-month formula:
Note: CME's published calculation yields 2.3350% due to slightly different day-counting conventions. The methodological principle remains identical.
Phase 3: Rate Change Calculation
Phase 4: Probability Extraction
Convert to 25bp units:
Decompose into characteristic and mantissa:
Extract binary probabilities:
The tree expands forward by repeating the process:
Starting point: \(\text{EFFR(Start)}_{\text{Nov}} = 3.0600\%\)
Following identical steps (details omitted for brevity), the CME methodology yielded:
The expanding tree generates four possible cumulative outcomes by November:
| Path | Sept Move | Nov Move | Cumulative | Probability |
|---|---|---|---|---|
| 1 | +50bp | +50bp | +100bp | 0.10 × 0.81 = 8.1% |
| 2 | +50bp | +75bp | +125bp | 0.10 × 0.19 = 1.9% |
| 3 | +75bp | +50bp | +125bp | 0.90 × 0.81 = 72.9% |
| 4 | +75bp | +75bp | +150bp | 0.90 × 0.19 = 17.1% |
Aggregating by cumulative change:
September 21, 2022: FOMC raised rates by 75bp (probability: 90%) ✓
November 2, 2022: FOMC raised rates by 75bp (conditional probability: 19% | Sept=75bp)
The methodology correctly identified the modal outcome for September but underestimated the probability of consecutive 75bp moves, illustrating that risk-neutral probabilities from futures may not perfectly match realized frequencies.
One of the most powerful features of the CME method is that it doesn't just predict one meeting - it can predict a whole sequence of meetings!
Today (Rate: 4.00%)
|
[Meeting 1]
/ \
+25bp (70%) Hold (30%)
/ \
Rate: 4.25% Rate: 4.00%
| |
[Meeting 2] [Meeting 2]
/ \ / \
+25bp (40%) Hold (60%) +25bp (50%) Hold (50%)
/ \ / \
4.50% 4.25% 4.25% 4.00%
Final probabilities:
- End at 4.50%: 70% × 40% = 28%
- End at 4.25%: (70% × 60%) + (30% × 50%) = 42% + 15% = 57%
- End at 4.00%: 30% × 50% = 15%
As you can see, the tree "expands" - each meeting doubles the number of possible paths!
With each additional Fed meeting, the possibilities multiply:
This is why computers are essential - the math gets complex very quickly.
The CME tool goes meeting by meeting, using the ending rate from one meeting as the starting rate for the next. It tracks all the paths and their probabilities, then shows you:
The expanding binary tree structure provides a systematic framework for tracking probability distributions across multiple sequential policy decisions.
Define state space at meeting \(t\):
For each state \(r_{t,i} \in \mathcal{S}_t\) with probability \(P_t(r_{t,i})\), the binary branching yields two possible successors:
Let \(p_{t,i}^{\uparrow}\) denote the probability of upward movement from state \(r_{t,i}\). The state probabilities at \(t+1\) aggregate from multiple paths:
where the transition probability \(p_{t,i}(r_{t,i} \to r)\) equals either \(p_{t,i}^{\uparrow}\) or \((1 - p_{t,i}^{\uparrow})\) depending on the branch.
The tree structure exhibits controlled combinatorial explosion:
However, many paths converge to the same terminal rate level, reducing the complexity of probability aggregation compared to tracking all paths individually.
The tree expansion can be represented as a state-transition system. Define probability vector:
And transition matrix \(\mathbf{T}_t\) where entry \(T_{ij}\) gives probability of transitioning from state \(i\) at meeting \(t\) to state \(j\) at meeting \(t+1\):
This matrix formulation enables efficient computation of forward probabilities and facilitates sensitivity analysis.
Multiple paths may lead to the same cumulative rate change. For example, after two meetings, a cumulative +50bp change can arise from:
The probability of ending at the target rate aggregates across all contributing paths:
Naive path enumeration requires \(O(2^T)\) operations for \(T\) meetings. However, dynamic programming reduces this to \(O(T^2)\) by aggregating probabilities at each state rather than tracking individual paths:
This algorithmic efficiency enables real-time calculation even for 8+ meeting horizons.
Zero Lower Bound: When rate approaches zero, upward branches continue normally but downward branches are constrained:
Rate Reversals: The binary assumption implicitly rules out immediate reversals (hike followed by cut or vice versa) within the near-term horizon. This reflects behavioral smoothing but may underestimate tail risks during policy uncertainty.
Non-Standard Increments: When futures imply moves larger than 25bp (characteristic ≥ 1), the tree structure accommodates this by treating larger moves as single branches rather than decomposing into multiple 25bp steps.
No forecasting method is perfect, and the CME expanding tree method has some known limitations. Understanding these helps you know when to trust the probabilities and when to be skeptical.
The method assumes 25bp moves. When the Fed does 50bp, 75bp, or emergency cuts, the binary tree structure has to adapt. It can handle this, but it's less elegant.
Example: March 2020 COVID emergency cuts between scheduled meetings
The binary tree says there are only two realistic options at each meeting. But what if markets are split three ways?
Example: Early 2023 when markets debated between: cut 25bp (30%), hold (40%), hike 25bp (30%)
The method would force this into two categories, distorting the true probability distribution.
Remember Assumption 7? Futures prices include a "risk premium" - traders pay extra for insurance. This means futures prices aren't pure predictions; they're slightly biased.
Research shows this bias is about 35-60 basis points per year, and it gets bigger during recessions.
The further out you go, the less reliable it gets:
This is because futures markets get less liquid the further out you go, and economic conditions can change dramatically.
The CME expanding tree method is an excellent tool for understanding short-term market expectations under normal conditions. But during crises, regime changes, or for long-term predictions, it should be combined with other methods like surveys, economic models, or expert judgment.
While the CME expanding tree methodology represents the industry standard for extracting policy expectations from futures, it embodies several structural limitations that constrain its domain of applicability.
The fundamental restriction to two outcomes per meeting node creates systematic distortions when genuine probability mass is distributed across three or more scenarios.
Mathematical Manifestation: Consider a situation where physical probabilities are:
The binary framework must force-fit this into two categories, resulting in:
where \(m\) is the mantissa. This necessarily misrepresents the true distribution, with the magnitude of distortion proportional to the probability mass on the excluded third outcome.
Consequences:
The methodology extracts risk-neutral (\(\mathbb{Q}\)) probabilities but policy forecasting requires physical (\(\mathbb{P}\)) probabilities. The wedge between these measures derives from risk premia:
Empirical Magnitudes (Piazzesi & Swanson 2008):
Failure to adjust for risk premia systematically biases probabilities:
The 25bp increment assumption, while historically justified, fails during crisis periods requiring aggressive policy action:
| Episode | Non-Standard Moves | Methodological Impact |
|---|---|---|
| 2001-2002 Recession | Multiple 50bp cuts | Binary tree adapts but loses elegance |
| 2008 Financial Crisis | 100bp cut (Oct), inter-meeting moves | Assumption 5 violated; probabilities unstable |
| 2020 COVID Crisis | 150bp emergency cut (March) | Extreme non-standard; futures-based forecasting breaks down |
| 2022-2023 Inflation Fight | Four consecutive 75bp hikes | Tree structure accommodates but underestimates consecutive large moves |
Forecast performance deteriorates systematically with horizon:
Drivers of Horizon Degradation:
Comparative Performance by Horizon (Gürkaynak et al. 2007):
The base CME methodology treats all rate changes symmetrically and independently. It does not model:
These behavioral and institutional features can be incorporated through enhanced frameworks (as discussed in our methodology), but they are absent from the baseline CME implementation.
Recommended Best Practices:
This page has provided a comprehensive deep dive into the CME expanding tree methodology. For information on how we adapt this methodology for the European Central Bank and Bank of England, return to the main methodology page.