# Fed Funds Rate Predictions: Assumptions

## Explanation of the Various Estimation Restrictions

### Single-Meeting Estimation

This technique is used to derive implied probabilities for the December and January FOMC meetings. The probabilities are estimated using a restricted least squares (RLS) technique explained in “Recovering Market Expectations of FOMC Rate Changes with Options on Federal Funds Futures ” by John B. Carlson, Ben Craig, and William R. Melick. The restriction is that the probabilities must sum to one. Sometimes, the RLS probabilities are not all nonnegative. In these cases, the probabilities are re-estimated with a nonnegativity constraint (subject to a low level of tolerance for constraint violations) using numerical techniques.

Typically the single meeting estimation technique is used whenever a FOMC meeting is not followed by another meeting in the following month. The probability density function (pdf) for January 2008 meeting outcomes can be estimated with the single meeting technique because there is no scheduled meeting in February. Under the assumption that there are no intermeeting changes, February options and futures prices only embed the probabilities of alternative January FOMC meeting outcomes. In all the January meeting models, there is the implicit assumption that the average monthly effective rate in February is expected to hit the target fed funds rate chosen on January 30.

After the June 25 meeting, implied probabilities for the August 5, 2008 meeting can be estimated with the single meeting estimation technique with August fed funds futures and options. In all of these models estimated with August options it is assumed that the expected average fed funds rate will equal the target fed funds rate chosen by the FOMC at the June 2008 meeting for any days in the August 1–August 5 period not in the past.

#### Futures Price Restriction (single-meeting estimation)

The probability-weighted sum of the alternative target rates must equal the futures rate derived from the settlement price of the fed funds futures contract for which the underlying options were used to do the estimation.

In some cases, the nonnegative probability constraint may not be able to be satisfied simultaneously with the futures constraint. In these cases, the nonnegativity constraint is dropped, and the probabilities are estimated with the RLS technique.

#### Term-Premium Restriction (single-meeting estimation)

The probability-weighted sum of the alternative target rates plus a one-basis-point premium for every 30 days until the futures contract of the underlying options expire must equal the futures rate derived from the settlement price of the futures contract. In some cases, the nonnegative probability constraint may not be able to be satisfied simultaneously with the term-premium constraint. In these cases, the nonnegatvity constraint is dropped, and the probabilities are estimated with the RLS technique.

### Joint-Meeting Estimation

This technique is used to derive implied probabilities prior to June 25 for the August 2008 FOMC meeting with July and August fed funds futures and options. Prior to June 25, joint meeting models were an alternative to the single meeting models used to derive implied probabilities for the August meeting with August fed funds futures and options. In the excel file and the slides the probabilities estimated with the joint meeting model are spliced together with the analogous single August meeting model estimated after June 25 with August options. The discussion below covers the period before June 25 when the June 2008 FOMC meeting target rate choice was not known.

In general the joint estimation technique is used to derive implied target rate probabilities for a meeting that is not the next meeting and that is followed by a meeting in the subsequent month. For the August meeting, implied probabilities for possible target rate paths are estimated for the June and August meetings with options written on the July futures contract and with options written on the August futures contract. An example of one “path probability” is the probability of the FOMC choosing a target fed funds rate of 2.00% at both the June and August meetings. From these path probabilities, the probabilities for possible target rates chosen at the second meeting of the path are obtained with straightforward arithmetic. The implied probabilities for this second meeting are the ones reported. The joint meeting models without a futures price or term-premium restriction are estimated using a restricted least squares (RLS) technique explained in “Recovering Market Expectations of FOMC Rate Changes with Options on Federal Funds Futures.” If any of the RLS estimates are negative, then the probabilities are re-estimated numerically with a nonnegativity constraint. The joint meeting models with futures price or term-premium restrictions (explained below) are always estimated numerically. In all models it is assumed that the daily effective rate is expected to hit the end of day target fed funds rate for each day in August. All joint meeting models have the restrictions that the path probabilities must be nonnegative and sum to one. In some cases the nonnegativity constraint will not be able to be satisfied simultaneously with one of the other restrictions. In these cases, the nonnegatvity constraint is dropped and the probabilities are re-estimated without this restriction.

#### Futures Price Restriction (joint-meeting estimation)

Given our assumption that market participants anticipate the effective fed funds rate to hit the target fed funds rate, an average monthly effective rate for both July 2008 and August 2008 can be computed for any given hypothetical target rate path for the June 24–25, 2008 meeting and the August 5, 2008 meeting. For example, for a hypothetical path where the FOMC chooses a target rate of 2.00% at the June meeting and a target rate of 2.25% at the August meeting, the monthly effective rate for July 2008 given our assumptions is 2.00% and the monthly effective rate for August 2008 is (5/31)*2.00% + (26/31)*2.25% = 2.209%. Call these the path implied July and August effective rates. For the futures restriction models, the “path probability” weighted sum of the alternative path implied July effective rates must equal the July fed funds futures rate, and the “path probability” weighted sum of the alternative path implied August effective rates must equal the August futures rate.

#### Term-Premium Restriction (joint-meeting estimation)

Using the same terminology as in the above paragraph, the joint models with a term-premium restriction have the following two restrictions. The “path probability” weighted sum of the alternative path implied July effective rates plus a one basis point premium for every thirty days between the estimation date and the expiration date of the July fed funds futures contract must equal the July fed funds futures rate, and the “path probability” weighted sum of the alternative path implied August effective rates plus a one basis point premium for every thirty days between the estimation date and the expiration date of the August fed funds futures contract must equal the August futures rate.