Intermeeting Rate Cuts as a Response to Rare Disasters

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Intermeeting Rate Cuts as a Response to Rare Disasters David S. Miller 2020-076 Please cite this paper as: Miller,DavidS.(2020). “IntermeetingRateCutsasaResponsetoRareDisasters,”Finance and Economics Discussion Series 2020-076. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.076. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Intermeeting Rate Cuts as a Response to Rare Disasters David S. Miller Federal Reserve Board∗ Abstract This paper measures the probability of rare disasters by measuring the probability of the intermeeting federal funds rate cuts they provoke. Differentiating between months with Federal Open Market Committee (FOMC) meetings and months without identifies excess returns on federal funds futures averaging -1.5 bps per horizon month-ahead at short horizons, corresponding to a 3-5% per month risk-neutral probability of an intermeeting rate cut. The excess returns differ between months with and without meetings, suggesting a positive risk premium associated with meetings. The federal funds excess returns explain a significant portion of equity excess returns, and hence the equity premium puzzle. JEL classification: E44, G12 Keywords: Rare Disasters, Equity Premium, Risk Premium, Federal Funds Futures ∗E-mail: david.s.miller@frb.gov. TheauthorthanksdiscussionswithChrisVickers,Francois Gourio, Jeff Campbell, Silvia Miranda-Agrippino, Jeff Huther, and Anthony Diercks. The views expressed in this paper are those of the author and not necessarily those of the Federal Reserve Board or of the Federal Reserve System.

1 Introduction Rare disasters, economic crises that are rarely observed but whose potential for economic destruction lays latent, have been put forth as an explanation for many economic phenomena, most prominently the equity premium puzzle as in Barro (2006). Federal funds futures offer a new way to identify the probability of such events: identify the probability of an emergency, unscheduled federal funds rate cut by the Federal Open Market Committee (FOMC) which would be brought on by a rare disaster. Excess returns on the futures are around -1.5 bps per contract month-ahead, translating to a risk-neutral probability of an intermeeting cut of 3-5% per month, similar to the empirical realization. Applied to the equity premium puzzle, the intermeeting cut probability strongly influences equity excess returns. I find the probability of an intermeeting rate cut using a simple identification scheme. Federal funds futures are available to predict the federal funds rate every month. However, the FOMC has scheduled meetings only 8 months of the year. Thus the federal funds futures covering the 4 remaining months shouldpredictthatthefederalfundsrateisunchangedoverthosemonths. The futures do not do so: during those 4 months, there is a statistically significant differenceof-1.5bps–theexcessreturns–betweentheratethefuturespredict and the actual federal funds rate, reflecting the probability of an unexpected intermeeting rate cut by the FOMC. In months with meetings, the difference is only -0.5 bps, suggesting the existence of a partially offsetting risk premium in those months. The excess returns have the opposite, positive, sign during the period when the federal funds rate was at the zero lower bound (ZLB). At the ZLB, further rate cuts, both expected and unexpected, were impossible. Any excess returns due to an unexpected cut disappear, leading to the positive excess returns. Moreover, during that period, and unlike normal times, there is no difference in excess returns between months with meetings and months without. An intermeeting rate cut is not the same as the usual definition of rare disaster: an intermeeting rate cut is a response to a rare disaster, rather than 1

the disaster itself. Historically, they have been induced by disasters such as the onset of recessions, banking crises, and a terrorist attack. For example, a sudden fall in the stock market is a rare disaster, while the intermeeting rate cut following the fall is a response to that fall, and could cause the stock market to rise, as happened in 2007. In this way, an intermeeting rate cut is a form of insurance, or Fed ‘put’, on the economy. As established by Lucca and Moench (2015), the Federal Reserve has significant power over equity returns. This relationship explains why I find that federal funds excess returns help explain equity excess returns: the more likely an intermeeting cut, the more exposed equities are to a disaster, even if the Fed intervenes. Federal funds futures can be used to generate a market-implied forecast of the federal funds rate. This forecast includes the possibility of intermeeting ratecuts. Usingthemeasuredexcessreturns,wecanadjustthemarket-implied forecast for the possibility of intermeeting cuts. This new market-implied forecast without intermeeting risk will better reflect the choices policymakers are expected to take if the economy continues to evolve along the expected path. To preview results, using 1 month ahead federal funds futures from 2000- 2019 excluding the ZLB period, I find excess returns of -1.6 bps for months without meetings, and excess returns of 0 for months with meetings. Using 2 month ahead federal funds futures, I find excess returns of -3.1 bps for contracts covering periods with 1 FOMC meeting, and excess returns of - 1.7 bps for contracts covering periods with 2 FOMC meetings. Regressions including the 1 month ahead excess returns, together with financial controls, can explain 25% of the variance of equity excess returns in months without meetings, and 10% of the variance if the after-meeting parts of months with meetings are included. This paper brings together literatures on excess returns in federal funds futures, rare disasters, and federal reserve impact on equity returns. The calculation of excess returns on federal funds futures has been done before, though not this paper’s identification method. Most notably, Piazzesi and T.Swanson (2008) find large excess returns using a sample from 1994-2006. 2

Bundick (2007) drops months with intermeeting moves and shows that excess returns are relatively small. This paper shows that excess returns are insignificant if the regressions do not control for whether a month has an FOMC meeting, and when that meeting occurs. Controlling for these factors reveals small, but significant, excess returns that correlate with intermeeting rate cut risk. The modern form of rare disasters was promulgated by Barro (2006). Rare extreme events can, in theory, explain much of the equity premium puzzle of Mehra and Prescott (1985). Gabaix (2012) formalizes the computation of the impact of rare disasters on various macro-finance puzzles. Barro and Ursua (2008) measures the risk of rare disasters by looking at cross-country consumption records. This paper offers a novel way to measure the probability of a specific type of rare disaster: the type of rare disaster that provokes an intermeeting cut by the FOMC. Other papers, such as Barro and Liao (2019), use options pricing to measure the risk priced into the stock market. A relatively new literature beginning with Lucca and Moench (2015) documents the impact the Federal Reserve has on equity markets. They find that a significant amount of equity returns are accumulated just after an FOMC meeting. Later work by Kurov et al. (2019) extends the sample of data used and finds this impact is disappearing. This paper finds a significant amount of equity returns is correlated with the probability of an intermeeting rate cut, hence is controlled by the Federal Reserve. The next section defines excess returns, and provides evidence of negative excess returns on federal funds futures. Section 3 shows the impact these excess returns have on equity excess returns. Section 4 shows how the excess returns correlate with other macro risk measures. 2 Excess Returns on Federal Funds Futures Federal funds futures are monthly contracts that settle at 100 minus the average effective federal funds rate for each month. For simplicity, I modify all reported contract prices to equal the average effective federal funds rate alone 3

rather than 100 minus the average effective federal funds rate. Define fn to be t the price of the federal funds future at the beginning of month t that covers the month n+t−1. r is the average effective federal funds rate over month t t. For example, f1 is the price of the front month contract on May 1st, May2008 2008, settling at r the average effective federal funds rate for May 2008. May2008 Define the excess return on a federal funds future of horizon n in month t as rxn = fn −r t+n t t+n While federal funds futures are futures, not forwards, treating them as forwards simplifies calculation and interpretation at shown in Piazzesi and T.Swanson (2008). The futures track the effective federal funds rate – the real-world federal funds rate faced by banks – rather than the target federal fundsrate,ormidpointoftherangeofthefederalfundsrate,setbytheFOMC. While the difference between these two measures was, at times, noisy before the ZLB, the two rates now move in parallel upon changes in the federal funds rate corridor. I will be running regressions of excess returns on indicators of how many monthswithFOMCmeetingsareintheexcessreturns’snhorizon,Meetings Indicator t+n and controls X t+n rxn = Meetings Indicator +X +(cid:15) t+n t+n t+n t+n When running these regressions, I include two controls to account for the state of financial markets and the difference between the target and effective federal funds rate. The first control is the level of the target federal funds rate, or midpoint of the federal funds rate range when applicable, from the beginning of the month. The second control is the standard deviation of the daily difference between the target federal funds rate and the effective federal funds rate, excluding days with meetings, from the previous month. Both of these controls are known by market participants at the beginning of the month. Inadditiontowhetheramonthhasameetingornot, monthswithmeetings 4

differ from each other based on when the meeting takes place. Regression controls include a variable “Days Left” that takes the value of the number of days after the meeting in a month with a meeting, and the value of the number of days in the month in a month without a meeting. See Figure 1. Underlying the definition of this variable is a hypothesis that the sections of months after a meeting differ from months without meetings only in the number of days. This hypothesis is confirmed in Appendix A. Figure 1: Days Left in Months with/without Meeting Month with meeting Days Left =15 Meeting 15th16th 1st 30th Month without meeting Days Left =30 1st 30th Futures data is from Bloomberg and covers Jan. 2000 - Aug. 2019. Almost all regressions will use data from Jan. 2000 - Aug. 2009 and Jan. 2016 - Aug. 2019, excluding months with intermeeting moves. I separate out contracts during the ZLB period (Jan. 2009 - Nov. 2015) due to the impossiblity of intermeeting cuts during this time. Unless otherwise noted, these months are not included in the results. The beginning and end of the ZLB period, Dec. 2008 and Dec. 2015 are dropped. I also drop Sept. 2008 - Nov. 2008 due to the unique stresses the federal funds market was under during this time. I remove contracts that cover months that had an intermeeting move in order to calculate the excess return in normal months Bundick (2007). Standard errors for horizons greater than 1 are heteroskedasticity and autocorrelation 5

consistent (HAC) due to overlapping contracts. Mean Excess Returns (Annualized bps) 2000-2019 Sample ZLB 1-month ahead −0.65 7.95∗∗∗ (3.41) (2.02) 2-months ahead 3.78 7.73∗∗ (4.72) (3.42) 3-months ahead 3.27 7.71∗∗ (6.52) (3.71) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 HAC standard errors for 2 and 3-month ahead contracts Table 1: Mean Excess Returns for 1, 2, and 3-month ahead contracts Table 1 presents the mean excess return on 1, 2, and 3-month ahead federal funds futures over the sample described above, and the ZLB period. In order to compare excess returns between contracts at different horizons, all results are in annualized basis points calculated by multiplying the excess returns by 12/n where n is the contract’s horizon. Notably, there are no statistically significant excess returns during the non-ZLB period, while the ZLB period has strong, positive, excess returns. 2.1 Results This paper’s main innovation is to identify the probability of an intermeeting rate cut by comparing months with an FOMC meeting from those without in 6

1-month ahead federal funds futures. ExcessReturns(Annualizedbps) (1) (2) MeetinginMonth 35.78∗∗∗ 33.74∗∗∗ (9.84) (9.89) DaysLeft 1.29∗∗∗ 1.21∗∗∗ (0.43) (0.43) Constant −49.32∗∗∗ −55.04∗∗∗ (14.08) (15.23) Controls No Yes Observations 142 140 AdjustedR2 0.08 0.10 FStatistic 6.80∗∗∗ (df=2;139) 4.94∗∗∗ (df=4;135) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01 Table 2: Excess Returns on 1-Month Ahead Futures Table 2 shows the main result. Months without meetings have significant negative excess returns. Using Column 2, and the mean number of days left in a month without a meeting, 30.1, excess returns in months without meetings average -18 annualized basis points, or -1.5 non-annualized basis points. Months with meetings average 13.8 days left, hence have excess returns of -5 annualized basis points, or -0.4 non-annualized basis points. Since all months have a chance of a disaster occurring during the month, they should all have similar excess returns.1. However, the difference in excess returns between months with meetings and months without is large, roughly 1 bp. Thesizeandsignofthisdifferencepointstotheexistenceofariskpremium associatedwithanFOMCmeeting, astheorizedbyMiranda-Agrippino(2016). Table 3 provides the same analysis for 2-month ahead futures. All the main results, extended to 2 months, hold. The number of meetings in the period covered by the contract is now either 1 or 2 meetings. (There are no 2 month 7

periods without a meeting.) The “2 Mo. Days Left” variable is the sum of the “Days Left” variable over the 2 months covered by the contract. ExcessReturns(Annualizedbps) (1) (2) Num. Meetings=2 22.17 17.12∗ (14.48) (9.81) 2Mo. DaysLeft 0.72 0.64∗ (0.55) (0.38) Constant −31.40 −44.70∗∗∗ (22.68) (16.41) Controls No Yes Observations 135 133 AdjustedR2 0.04 0.18 FStatistic 3.68∗∗ (df=2;132) 8.42∗∗∗ (df=4;128) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01 HACstandarderrors Table 3: Excess Returns on 2-Month Ahead Futures Using Column 2, months with 1 meeting average 43 days left over the contract period, hence excess returns are -16.5 annualized basis points, or - 2.75 non-annualized basis points. Months with 2 meetings average 29.8 days left over the contract period, hence excess returns are -7.9 annualized basis points, or -1.3 non-annualized basis points. The existence and size of excess returns at the 2-month horizon confirms the intuition behind the 1-month ahead excess returns. The probability of an intermeeting rate cut over the next two months should be strictly larger than the probability of an intermeeting rate cut over the next month. The actual size, roughly 2x, fits closely with the 1-month ahead excess returns findings. Thedifferenceinexcessreturnsbetweencontractscovering1meeting vs. contracts covering 2 meetings shows the risk premium associated with 8

meetings continues. See Appendix B for the equivalent table for 3-month ahead futures. They do not show any statistically significant relationships. The number of independent observations decreases with longer horizons as more contracts cover months with intermeeting moves and hence are removed from the sample. Fewer observations combined with the impact of using HAC standard errors lead to no significance. 2.1.1 Expected Federal Funds Path Without Intermeeting Risk We can use the excess return results to produce a simple forecast of monetary policy purged of the possibility of intermeeting cuts. Such a forecast would represent the market-implied belief about the choices of the FOMC at its regularly scheduled meetings. To produce the forecast, simply add 1.5 bps per contract-month ahead to the market price of the futures contract. For example, 1-month ahead contracts would be adjusted by 1.5 bps, and 3-month ahead contracts by 4.5 bps. The scale of adjustment of the federal funds rate forecast is very similar to the adjustment derived through measurement of the risk premium. Diercks and Carl (2019) using the covariance of real variables, find a risk premium on federal funds futures of around -1 bps per month. The similarity suggests that the majority of the risk in the market risk premium is the risk of an intermeeting rate cut. 2.1.2 Intermeeting Risk Probability Translating the excess returns into the probability of a rate cut per month requiresassumptionsaboutthesizeofanintermeetingratecut,andcorrelation of risks. I assume any cut will be 50 bps, as almost all modern cuts have been. Define p do be the per month cut probability, and p to be the daily m d 1A disaster occurring in a month with a meeting would not necessarily provoke an intermeeting ratecut. Instead,thedisasterwouldforcealargercutatthescheduledmeeting than demanded by the non-disaster economic conditions foreseen at the beginning of the month. 9

probability. I analyze the two extreme cases: rate cuts are either a once per month, or once per day calculation. First, assumethattheriskofanintermeetingcutisdependentoneconomic conditions, and thus doesn’t change much over a month. This calculation is simple: (−50)p = −1.5 hence the probability is 3%. Second, assume that the m risk of an intermeeting cut is identical and independent per day, dependent on unpredictable events such as terrorist attacks. After an intermeeting rate cut, the rate stays cut for the rest of the month, hence the probability of a lower rate on any day depends non-linearly on the number of days that came before it. Figure 2 illustrates a two-day example where the rate either stays constant, or is cut by 50 bps. If each day has an independent probability p of a cut, the price of the contract, and hence the d excess returns, has a p2 term. d Figure 2: Non-Linear Impact of Days Left 1−p p Expected Rate Day d d 1st 100 50 (1−p d )(100)+p d (50) 1−p d p d 1 2nd 100 50 50 (1−p d )2100+(1−p d )p d (50)+p d (50) Price= 1(cid:2) (1−p )(100)+p (50)+(1−p )2100+(1−p )p (50)+p (50) (cid:3) 2 d d d d d d Then using the same logic as Figure 2 that each day has a probability of a cut, and the probability that there was a cut on a previous day, (cid:32) (cid:33) 30 1 (cid:88) (31−t)(1−p )t−1p (−50) = −1.5 d d 30 t=1 1−(1−p )30 = p d m Solving for p shows there is a roughly 5.5% chance of an intermeeting rate m 10

cut per month. 3 Equity Returns Iusethemeasureoffederalfundsexcessreturnstoseetheimpactintermeeting risk has on equity excess returns. The data for equity excess returns comes fromKenFrench’swebsite. Intermeetingriskisameasureofdisasterrisk. The higher the probability of an intermeeting cut, the higher the risk of disaster. As equities are risky assets, their return should be higher at times of higher risk. EquityExcessReturns Non-MeetingMonths AfterandNon-MeetingMonths (1) (2) (3) (4) ExcessReturns −0.05∗∗ −0.07∗∗∗ (0.03) (0.02) Alt. ExcessReturns −0.03∗∗ −0.02∗ (0.01) (0.01) DaysLeft 0.07∗ 0.07∗ (0.03) (0.04) MeetinginMonth 2.55∗∗∗ 2.43∗∗∗ (0.75) (0.74) Constant −1.39∗∗ −0.02 −3.18∗∗∗ −2.18∗ (0.61) (0.91) (1.14) (1.21) Controls No Yes No Yes Observations 47 40 134 132 AdjustedR2 0.06 0.24 0.08 0.11 FStatistic 4.19∗∗ 5.14∗∗∗ 4.60∗∗∗ 4.20∗∗∗ (df=1;45) (df=3;36) (df=3;130) (df=5;126) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01 Table 4: Equity Excess Returns 11

Table4showstheimpactfederalfundsexcessreturnshaveonequityexcess returns. Columns1and2useequityexcessreturndataonlyfrommonthswithout meetings. The negative coefficients indicate that negative federal funds excess returns are connected to higher equity excess returns. The tail risk represented by intermeeting rate cut risk has a significant impact on equity excess returns in non-meeting months. In order to incorporate more data, I define the Alternative Excess Returns variable as the excess returns that exist in the parts of months after meetings in months with meetings, and the excess returns from the months without meetings. Figure 3: Partial Month with Meeting and Whole Month without Meeting Month with meeting Days Left =15 Alt. ER=Price −Payoff Meeting 16 15th16th 1st 30th Month without meeting Days Left =30 Alt. ER=Price −Payoff 1 1st 30th Columns 3 and 4 use these alternative excess returns to show their impact on equity excess returns. While relationship is not as strong as when using solely the non-meeting months, the relationship is still statistically significant. 4 Other Risk Measures InthissectionIshowhowexcessreturnsonfederalfundsfuturesareconnected with other measures of market risk. Specifically, I relate excess returns to 12

the VIX measure, the excess bond premium of Favara et al. (2016), the rare disaster probability derived from S&P options computed by Barro and Liao (2019), and the economic policy uncertainty index of Baker et al. (2016). VIX EBP Rare Disaster EPU Excess Returns −0.10∗∗ −0.01 −0.001∗∗ 0.001 (0.04) (0.004) (0.0002) (0.16) Constant Yes Yes Yes Yes Controls Yes Yes Yes Yes Observations 41 41 36 41 Adjusted R2 0.18 0.08 0.17 0.22 F Statistic 3.97∗∗ (df = 3; 37) 2.21 (df = 3; 37) 3.36∗∗ (df = 3; 32) 4.66∗∗∗ (df = 3; 37) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Table 5: Relationship between Excess Returns and Other Risk Measures in Non-Meeting Months I present the results as regressions in Table 5 in order to follow the conventions earlier, and include the same financial controls. As Table 5 shows, excess returns are highly correlated with both the VIX Index and the Rare Disaster measure. These relationships are of the proper sign to add evidence that excess returns on federal funds futures are a measure of rare disaster risk, albeit a rare disaster that induces an intermeeting rate cut. 5 Conclusion It’s not easy to define a disaster, but federal funds futures offer a way to do so. A disaster is any negative event that causes an intermeeting rate cut. Federal funds futures reflect this possibility, leading to negative excess returns. The risk of such disasters helps explain a significant portion of equity excess returns, confirming that disasters are important to asset pricing. References Baker, Bloom, and Davis, Measuring Economic Policy Uncertainty 2016. 13

Barro, Robert and J.F. Ursua, “MacroeconomicCrisessince1870,” Brookings Papers on Economic Activity, 2008, Spring. Barro, Robert J., “Rare Disasters and Asset Markets in the Twentieth Century,” Quarterly Journal of Economics, 2006, 121 (3). and Gordon Y. Liao, “Tractable Rare Disaster Probability and Options- Pricing,” FEDS Notes. Board of Governors of the Federal Reserve System, 2019. Bundick, Brent, “Risk-Adjusted Futures and Intermeeting Moves,” Federal Reserve Bank of Kansas City Working Paper, June 2007. Diercks, Anthony M and Uri Carl, “A Simple Macro-Finance Measure of Risk Premia in Fed Funds Futures,” FEDS Notes. Board of Governors of the Federal Reserve System, 2019. Favara, Giovanni, Simon Gilchrist, Kurt F. Lewis, and Egon Zakrajsek, Recession Risk and the Excess Bond Premium, Board of Governors of the Federal Reserve System, 2016. Gabaix, Xavier, “Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance,” Quarterly Journal of Economics, May 2012, 127 (2). Kurov, Alexander, Marketa Wolfe, and Thomas Gilbert, “The Disappearing Pre-FOMC Announcement Drift,” SSRN, 2019. Lucca, David O. and Emanuel Moench, “The PreFOMC Announcement Drift,” Journal of Finance, 2015, January. Mehra, R and E C Prescott, “The equity premium: a puzzle,” Journal of Monetary Economics, 1985, 15 (2). Miranda-Agrippino, Silvia, “Unsurprising Shocks: Information, Premia, and the Monetary Transmission,” Centre for Macroeconomics DP 2016-13, 2016. 14

Piazzesi, Monika and Eric T.Swanson, “Futures prices as risk-adjusted forecasts of monetary policy,” Journal of Monetary Economics, 2008, 55 (4). 15

A Months of Unusual Size In addition to whether a month has a meeting or not, months with meetings differ from each other based on when the meeting takes place. Regressions will include a variable “Days Left” that takes the value of the number of days after the meeting in a month with a meeting, and the value of the number of days in the month in a month without a meeting. Underlying the definition of this variable is a hypothesis that the sections of months after a meeting differ from months without meetings only in the number of days. Figure 4: Synthetic Month and Whole Month without Meeting Month with meeting Days Left =15 Alt. ER=Price −Payoff Meeting 16 15th16th 1st 30th Month without meeting Days Left =30 Alt. ER=Price −Payoff 1 1st 30th To test this hypothesis, I create synthetic months, calculating alternative excess returns on the 1-month ahead contract for the partial month after meetings. If the meeting occurs on the last day of the month, I drop the month from the sample. See Figure 4 for a visual explanation of the variables. Combining these synthetic months of varying lengths with the set of whole months without meetings, I regress the alternative excess returns on the number of days in the whole or synthetic month. 16

Alt. ExcessReturns(Annualizedbps) (1) (2) (3) (4) DaysLeft −0.28∗ −0.26 (0.16) (0.17) (DaysLeft)2 −0.01∗ −0.01∗ (0.004) (0.005) Constant 1.51 −1.18 0.005 −2.55 (3.70) (4.60) (2.83) (3.87) Controls No Yes No Yes Observations 134 132 134 132 AdjustedR2 0.01 0.01 0.02 0.01 FStatistic 3.01∗ (df=1;132) 1.25(df=3;128) 3.38∗ (df=1;132) 1.37(df=3;128) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01 Table 6: Impact of Days Left on Excess Returns Table 6 shows the hypothesized relationship between the number of days in the month fragment and excess returns. Columns 1 and 2 provide evidence that each day in the fragment results in -0.3 annualized basis points in excess returns. Figure 5: Non-Linear Impact of Days Left 1−p p Expected Rate Day d d 1st 100 50 (1−p d )(100)+p d (50) 1−p d p d 1 2nd 100 50 50 (1−p d )2100+(1−p d )p d (50)+p d (50) Price= 1(cid:2) (1−p )(100)+p (50)+(1−p )2100+(1−p )p (50)+p (50) (cid:3) 2 d d d d d d 17

Columns3and4provideevidenceofafurtherrefinementofthehypothesis: alternativeexcessreturnsshouldbenon-linearinthenumberofdaysleft. After an intermeeting rate cut, the rate stays cut for the rest of the month. Hence the probability of a lower rate on any day depends non-linearly on the number of days that came before it. Figure 5 illustrates a two-day example where the rate either stays constant, or is cut by 50 bps. If each day has an independent probability p of a cut, the price of the contract, and hence the excess returns, d has a p2 term. In general, most regressions will show a better fit when Days d Left is taken to a power greater than 1. The importance of the number of days left also provides an opportunity to illustrate the difference between times when the federal funds rate was above the ZLB, and the ZLB period. Table 7 runs a similar regression to Table 6, but the sample now includes ZLB months. An indicator variable, interacted with the number of days left, denotes whether a month was during the ZLB period or not. As both columns show, each day left results in negative excess returns during normal times when an intermeeting rate cut was possible, and positive excess returns during the ZLB period when an intermeeting rate cut was impossible. 18

Alt. Excess Returns (Annualized bps) (1) (2) Days Left x (ZLB = 0) −0.29∗∗ (0.12) Days Left x (ZLB = 1) 0.28∗∗ (0.14) Days Left2 x (ZLB = 0) −0.01∗∗∗ (0.003) Days Left2 x (ZLB = 1) 0.01∗∗ (0.004) Constant −0.05 0.03 (2.66) (2.17) Controls No No Observations 212 212 Adjusted R2 0.08 0.08 F Statistic (df = 4; 207) 5.86∗∗∗ 5.84∗∗∗ Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Table 7: Impact of Days Left During ZLB B 3-month Ahead Futures Repeating the same analysis that was done with 1 and 2-month ahead futures shows nothing at the 3-month ahead horizon. A 3 month contract can cover 19

a period with 1, 2, or 3 meetings. I add up the Days Left variable in each of the 3 months covered to produce “3 Mo. Days Left”. Excess Returns (Annualized bps) (1) (2) Num. Meetings = 2 3.19 2.19 (10.85) (11.17) Num. Meetings = 3 34.07 23.22 (32.62) (22.45) 3 Mo. Days Left 0.04 −0.001 (0.42) (0.32) Constant −4.84 −16.76 (32.93) (26.55) Controls No Yes Observations 128 126 Adjusted R2 0.02 0.16 F Statistic 1.71 (df = 3; 124) 5.67∗∗∗ (df = 5; 120) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Table 8: 20