# How does one estimate the probability of the Fed increasing its benchmark rate based on Fed funds futures?

Fed funds futures show a 67 percent chance the central bank will increase its benchmark rate by year-end from virtually zero, according to data compiled by Bloomberg. The central bank last raised the rate in 2006.

http://www.bloomberg.com/news/articles/2015-07-16/dollar-bulls-accelerate-charge-as-yellen-provides-rates-spur

I am not sure how that probability was computed. However, the standard approach is to use fed futures to proxy for the "unexpected change" of FED rate. The most prominent reference is Bernanke and Kuttner (2005).

What they do, is to estimate the unexpected FED target rate change by doing: $$\Delta i^u = \frac{D}{D-d}(f_{m,d}^0-f_{m,d-1}^0)$$

where $$f_{m,d}^0$$ is the current month future rate and $$D$$ is the number of days in the month.

The extension to probability of change is given in Geraty (2000).

Where he basically estimates the probability $$p$$ of change by doing: $$p=\frac{\text{Fed funds rate implied by futures contract} - \text{The current fed funds rate}}{\text{Fed funds rate assuming a rate hike} - \text{The current fed funds rate}}$$

• The one from Geraty (2000) is the easiest to understand. Thanks. Oct 28 '15 at 4:51

There is actually a lot of art involved. The most simplistic framework is as follows:

The first step is to obtain a list of FOMC meeting dates. These are available currently for 2015 and 2016 here. If you're interested for rate expectations beyond 2016, you'd need to "guess" the meeting dates in the future based on past patterns.

The next step is to calculate the implied target rate coming out of each meeting. This is best explained with an example. Let's assume that the current Fed funds target rate is 0.125% (it's actually a range of 0%-0.25%) and that the next FOMC meeting is on September 17th, 2015 (actually July, but let's stay simple). The September 2015 Fed Funds future price is 99.835, implying a rate of $100-99.835 = 0.165\%$. Recall that Fed funds futures price is based on the arithmetic average of the daily Fed Funds effective rate, so we must have $$0.165\% = \frac{0.125\% \times 17 + r \times 13}{30},$$ where $r$ is the target rate coming out of the September meeting. (The first 17 days are based on rate coming out of the previous meeting, and the remaining 13 days based on the rate after the September meeting.) This gets us $r = 0.217307692\%$.

Finally, we can compute the probability of a rate hike. The assumption we'll use is that the Fed will either raise rate by 25bp or keep it unchanged. Assuming the probability of a 25bp hike is $p$, then we must have $$0.217307692\% = p\times 0.375\% + (1-p) \times 0.125\%.$$ (probability-weighted average of keeping rates at 0.125% or raising rates 25bp to 0.375%). This allows you to solve for $p$.

The example above is a gross simplification. More sophisticated models would adjust for term premium (since observed interest rate is not equal to rate expectations), allow for more scenarios coming out of each meeting, etc. If you're interested in longer-dated rate expectations, Eurodollar futures must be used as well.

Here are some useful references:

• Lehman Bros report: B. Tuckman and D. Calistru “The FEDISCOPE,”, September, 2003. Could not find a public copy, unfortunately. Jul 2 '19 at 0:27
• @AlexC Ah they filed a patent: patents.google.com/patent/US20060080203. Basically the same report. Jul 2 '19 at 0:29
– Attack68
Aug 2 '19 at 6:09
• @Attack68 Much appreciated! Aug 5 '19 at 3:20

Extending @Helin answer it is impossible to uniquely pin down more than two assumed scenarios due to limited information.

Suppose the current central bank deposit rate is 0.2% and the OIS/fed funds market implies that after the next central bank meeting the rate will be 0.35%, then there might be many different scenarios each with a probability, $$p_i$$;

1. A no change outcome
2. A 10bps hike to 0.3%.
3. A 20bps hike to 0.4%.
4. A 30bps hike to 0.5%.

You only have 2 equations for 4 unknowns:

1. All probabilities sum to 1.
2. The expected hike is 0.15ps (equivalent to the market price of 0.35%).

For example a valid solution is all probabilities are 25%, or indeed another valid response is {0%, 50%, 50%, 0%}.

To get around this you might introduce some smoothing or interpolation of probabilities to ensure that your end results is intuitively realistic.