Create Calendar for US Federal Holidays

We will need to calculate the number of business days around each FOMC meeting. Thus, a holiday calendar is essential to perform these calculations. The bizdays library is used to perform this.

# Load the US Holidays from a CSV file which I manually cleansed
# by comparing US holidays in timeanddate.com to a dataset in Kaggle.com
# which had errors.
#-----------------------------------------------------------------
us_holidays = read_csv("usholidays_cleansed.csv")
us_holidays %>% mutate( holiday = mdy(Date)) -> us_holidays
str(us_holidays)

## Classes 'spec_tbl_df', 'tbl_df', 'tbl' and 'data.frame': 484 obs. of  4 variables:
##  $ Index  : num  1 2 3 4 5 6 7 8 9 10 ...
##  $ Date   : chr  "1/1/1966" "2/22/1966" "7/4/1966" "9/5/1966" ...
##  $ Holiday: chr  "New Year’s Day" "Washington’s Birthday" "Independence Day" "Labor Day" ...
##  $ holiday: Date, format: "1966-01-01" "1966-02-22" ...

#
# Define the US observed federal holidays calendar from 1966-2020
# ---------------------------------------------------------------
mycal = create.calendar(name='MyCalendar', holidays=us_holidays$holiday, weekdays=c('sunday','saturday') , adjust.from = adjust.next , adjust.to=adjust.previous)

Identify the FOMC cycle time of each date

There are 8 FOMC meetings in a typical year. For each FOMC date, we need to find all business dates within a period before and after that date.

An FOMC date is defined as the last of 2 meeting dates when the FOMC meets to discuss and announce the target policy federal funds rate. At 2:00 or 2:15pm ET on the FOMC date, the statement is released to the public. Financial market practitioners are waiting to hear the outcomes.

For FOMC cycle purposes, the FOMC date is labelled as day 0 in its cycle.

Day 1 is the first business day after the FOMC date. Day 2 is the second business day afterwards and so forth. Likewise, day -1 is the first business date before the FOMC date.

We display the list of boundaries defining the weeks of an FOMC cycle expressed in business day offsets from the FOMC date. Each week below includes both its boundary dates.

weekDef = data.frame( weekNum = c(-1, 0, 1, 2, 3, 4, 5, 6) ,
                 start = c( -6, -1, 4, 9, 14, 19, 24, 29) ,
                 end = c( -2, 3, 8, 13, 18, 23, 28, 33)
                 )

weekDef %>% kable() %>% kable_styling() 

fomc_dates = read_csv("fomc_dates_1994_2019.csv")

fomc_dates %>% mutate( mdyDate = mdy(Date)) -> fomc_dates

Load the Equity Total Return Index

Use the Russell 1000 priced dates as the total set of business days and convert them to date objects. Note that we need prices for FOMC dates from 1994-02-04 to 2019-03-20. Based on this constraint of FOMC dates, we will chose equity prices from a date range relevant to the FOMC cycle calculations of this dates. This requires equity prices in a range from Jan 27, 1994 to May 10, 2019. The basis for this date range will be fully explored later when discussing FOMC cycle time.

#
#  Reads the Russell 1000 Date and Index Level.
# -----------------------------------------------------------------------
raw_russell = read_csv("RU1000TR.csv", col_types = cols( RU1000TR = col_double() ) )

# Convert the raw date column into a Date object and filter on the relevant range + a slight extra days in Jan 1994.
# ------------------------------------------------------------------------------------------------------------------
raw_russell %>% mutate( mdyDate = mdy(DATE)) %>%
           filter( (mdyDate >= "1994-01-04") & (mdyDate <= "2019-05-10") ) -> russ

ggplot(data=russ) + geom_point( aes(x = mdyDate , y = russ$RU1000TR) ) + ggtitle("Russell 1000 Index", subtitle="Prices from 1994-2019")

## Load the Treasury-Bill Yields

To calculate the excess return of equities over the risk free rate, we also need to obtain a proxy for the risk free rate. The best proxy that I found was the US Treasury Bill 3 month rate. The requirements are to have daily data from 1994 to 2019.
Other time series like the 1 month US Treasury Bill rate do not have adequate time series history.

In the code below, I download the 3 month T-Bill rate from the Federal Reserve’s H15 Report of interest rate data. Next, I parse the file, skipping header rows and extracting the relevant two columns: the date and rate.

There are several challenges which we overcome: + Header section occupies the first 5 rows + file contains 3m month rates in the third column + missing values occur on federal holidays with a placeholder string “ND” + column names are inappropriate + column datatypes need to be transformed into dates and doubles for dates and rates respectively.

#But these transformations are incomplete because holidays are displayed in the file showing a rate value "ND" for the holiday date row.  These #need to be removed in order for the rate data to be suitably convertible to a numerical type.

FRB_H15_data = read_csv("FRB_H15.csv", skip=5) %>% 
  select( c(1, 3))  %>% 
  rename( date = `Time Period`, rate = `RIFLGFCM03_N.B`) %>% 
  filter( !(rate=="ND")) %>%
  mutate( rate = as.double(rate)) -> tbill3m

Joining Equity Prices with T-Bill Rates is necessary to calculate the excess return for each observation.

russ %>% left_join( tbill3m, by = c( "mdyDate" = "date") ) -> eq_data

#  Replace missing T-Bill rates with the prior day's non-missing T-Bill rate 
# ---------------------------------------------------------------------------
eq_data$rate = na.locf(eq_data$rate)  # zoo library provided a known function

Calculate the Equity Premium

Our observations of the equity premium will range from Feb 2, 1994 to May 6, 2019. However, since the equity premium is calculated according over a 5-day window spanning the business day prior to the observation date to 4 days after, we need equity and yield date from Jan 27, 1994 to May 10, 2019. Later this data will be truncated to avoid windowing effects, but that will not impact statistical significance.

The equity premium \(h(t)\) is defined as for observation date \(t\) as:

\[h(t) = \left[ \frac{S(c(t,4))}{S(c(t,-1))} -1 \right] - \frac{c(t,4)-c(t,-1)}{30}\times (\frac{Y(t)}{100} \frac{91}{365})\]

Here \(c(t,d)\) represents the business date which is d days offset from date \(t\). Hence \(c(t,4)\) is 4 business days after \(t\). Also, \(Y(t)\) defines the 3-month Treasury bill rate expressed as percent in the raw file. Lastly, \(S(t)\) represents the equity index level on date \(t\).

We illustrate the first 100 rows in the table below.

# Use a recipe to conditionally mutate a row within the interior of the time period in the dataframe.
#---------------------------------------------------------------------------------------------------------

eq_data %>% mutate( equity_lead = lead(RU1000TR, 4, default=NA),
                      equity_lag = lag(RU1000TR, 1, default=NA) ,
                      days_lag    = lag(mdyDate, 1),  
                      days_lead   = lead(mdyDate, 4 ) ,
                      days_diff = as.numeric(days_lead - days_lag, units="days") ,
                      excess_ret = (equity_lead/ equity_lag - 1.0) - 0.01 * (days_diff)/30 * (rate/100) *(91/365)
                      ) -> eq_data
str(eq_data)

## Classes 'spec_tbl_df', 'tbl_df', 'tbl' and 'data.frame': 6398 obs. of  10 variables:
##  $ DATE       : chr  "1/4/1994" "1/5/1994" "1/6/1994" "1/7/1994" ...
##  $ RU1000TR   : num  883 884 883 888 897 ...
##  $ mdyDate    : Date, format: "1994-01-04" "1994-01-05" ...
##  $ rate       : num  3.15 3.14 3.12 3.07 3.06 3.04 3.02 3.01 3.03 3.03 ...
##  $ equity_lead: num  897 896 896 894 898 ...
##  $ equity_lag : num  NA 883 884 883 888 ...
##  $ days_lag   : Date, format: NA "1994-01-04" ...
##  $ days_lead  : Date, format: "1994-01-10" "1994-01-11" ...
##  $ days_diff  : num  NA 7 7 7 7 7 7 7 7 7 ...
##  $ excess_ret : num  NA 0.0146 0.0129 0.012 0.0111 ...

eq_data[1:100,] %>% kable() %>% kable_styling() %>% scroll_box(height = "400px", width="100%")

Calculating FOMC Cycle Dates

N = length(eq_data$mdyDate)
M = length(fomc_dates$mdyDate)

# Store the FOMC Cycle information into the data frame.
# We store a lower FOMC date.  I.e. the greatest date less than trade date
#    upper FOMC date is the least FOMC date greater than trade date.
#    When no lower FOMC date exists because trade date is before first FOMC date, we set lowerIndex =0
#    When no upper FOMC date exists because trade date is after last FOMC date, we set upperIndex = M + 1
#
#    offsetL is the number of business days from the lower FOMC date to the trade date.  It is non-negative.
#
#    offsetU is the negative of number of business days from upper FOMC date to trade date. It is non-positive.
#
#    weekNum is a number from 0,1-6 for the number of 
# ----------------------------------------------------------------------------------------------------
cycle_data = data.frame( trade_date = eq_data$mdyDate, 
                         lowerIndex = rep(0,N),
                         upperIndex = rep(0,N),
                         lowerFOMC  = as.Date(integer(N)) ,
                         upperFOMC  = as.Date(integer(N)) ,
                         offsetL    = rep(0,N),
                         offsetU    = rep(0,N),
                         weekNum    = rep(0,N),
                         dum_w0     = rep(0,N),
                         dum_w2     = rep(0,N),
                         dum_w4     = rep(0,N),
                         dum_w6     = rep(0,N),
                         dum_w0246  = rep(0,N),
                         dum_w246   = rep(0,N)
                         )

# Located the lower FOMC date by a linear search algorithm that re-uses the prior matches'
# FOMC date location to find the correct FOMC date for the next trade date.
# Bootstrap:  Identify the lower FOMC date for the earliest trade dates by searching from
#             right to left backwards in time.
# --------------------------------------------------

lowerIndices = findInterval(as.integer(eq_data$mdyDate), as.integer(fomc_dates$mdyDate))

for(i in 1:N)
{
    if( lowerIndices[i]== 0 )
    {
         cycle_data$lowerFOMC[i] = as.Date( mdy("01-01-1970") )
    }
    else
    {
         cycle_data$lowerFOMC[i] = fomc_dates$mdyDate[lowerIndices[i]]
    }
}

cycle_data$lowerIndex = lowerIndices

# Populate the offsetL = the number of business days AFTER the lowerFOMC date
# This is used to calibrate the business cycle and ultimately the dummy variables
# in the regression analysis
# -----------------------------------------------------------------------------
for(i in 1:N)
{
     if( cycle_data$lowerIndex[i] == 0 )
     {
          cycle_data$offsetL[i] = NA
     }
     else
     {
          cycle_data$offsetL[i] = bizdays(cycle_data$lowerFOMC[i], cycle_data$trade_date[i], mycal)
     }
}

# Populate the upper FOMC date offset. This will be a negative number
# since it counts the number of days BEFORE the FOMC date.
# ----------------------------------------------------------------
for(i in 1:N)
{
    # If a trade date has no lower FOMC date 
    # then its first upper FOMC date must be the first one.
    #----------------------------------------
    if( cycle_data$lowerIndex[i] == 0 )
    {
        cycle_data$upperIndex[i] = 1  
    }
    else
    {
        if(cycle_data$lowerIndex[i] > 0 && fomc_dates$mdyDate[ cycle_data$lowerIndex[i] ] == eq_data$mdyDate[i] )
        {
            cycle_data$upperIndex[i] = cycle_data$lowerIndex[i]  # This is an FOMC date
        }
        else
        {
            # fomc_dates$mdyDate[ cycle_datalowerIndex[i] ] < eq_data$mdyData[i] 
        
            cycle_data$upperIndex[i] = cycle_data$lowerIndex[i] + 1
        }
    }
}

for(i in 1:N)
{
    if( cycle_data$upperIndex[i] <= M )
    {
        cycle_data$upperFOMC[i] = fomc_dates$mdyDate[ cycle_data$upperIndex[i] ]
    }
}

#
# Populate the negative FOMC cycle offsets
# -----------------------------------------------------------
for(i in 1:N)
{
     if( cycle_data$upperIndex[i] > M )
     {
          cycle_data$offsetU[i] = NA
     }
     else
     {
          cycle_data$offsetU[i] = -bizdays(cycle_data$trade_date[i],cycle_data$upperFOMC[i], mycal)
     }
}     

Finally, we have to classification on each trade date to belong to one of weeks 0,1,2..6. Except for day -1 of an FOMC cycle which is identified by a value of \(offsetU = -1\), all other days of an FOMC cycle are classified by the value of \(offsetL\) based on the earlier table defining the week ranges.

for(i in 1:N)
{
    if( !is.na(cycle_data$offsetU[i]) && cycle_data$offsetU[i] == -1)
    {
        cycle_data$weekNum[i] = 0
    }
    else
    {
        z = cycle_data$offsetL[i]
      
        if(is.na(z))
        {
            cycle_data$weekNum[i] = NA    
            next;
        }
        
        if( (z >= 0) && (z <= 3))
        {
            cycle_data$weekNum[i] = 0
        }
        if( z >= 4 && z <= 8)
        {
            cycle_data$weekNum[i] = 1
        }
        if( z >= 9 && z <= 13)
        {
            cycle_data$weekNum[i] = 2 
        }
        if(z >= 14 && z <= 18 )
        {
            cycle_data$weekNum[i] = 3
        }
        if(z >= 19 && z <= 23)
        {
            cycle_data$weekNum[i] = 4
        }
        if(z >= 24 && z <= 28)
        {
            cycle_data$weekNum[i] = 5
        }
        if(z >= 29 && z <= 33)
        {
            cycle_data$weekNum[i] = 6
        }
        if(z >= 34 && z <= 38)
        {
            cycle_data$weekNum[i] = 7
        }
        if(z >= 39 && z <= 43)
        {
            cycle_data$weekNum[i] = 8
        }
    }
}

for(i in 1:N)
{
    if( is.na(cycle_data$weekNum[i]) )
    {
        next;  
    }
    else
    {
        weekNum = cycle_data$weekNum[i]
        
        # default is false
        cycle_data$dum_w0[i] = 0
        cycle_data$dum_w0246[i] = 0
        cycle_data$dum_w2[i] = 0
        cycle_data$dum_w4[i] = 0
        cycle_data$dum_w6[i] = 0
        cycle_data$dum_w246[i] = 0
        
        if( weekNum == 0)
        {
            cycle_data$dum_w0[i] = 1
            cycle_data$dum_w0246[i] = 1
        }
        if( weekNum == 2)
        {
            cycle_data$dum_w2[i] = 1
            cycle_data$dum_w246[i] = 1
            cycle_data$dum_w0246[i] = 1
        }
        if( weekNum == 4 )
        {
            cycle_data$dum_w4[i] = 1
            cycle_data$dum_w246[i] = 1
            cycle_data$dum_w0246[i] = 1
        }
        if( weekNum == 6 )
        {
            cycle_data$dum_w6[i] = 1
            cycle_data$dum_w246[i] = 1
            cycle_data$dum_w0246[i] = 1
        }
    }
}

Finally we display the fully prepared FOMC cycle information. For each potential observation we have calculated the associated lower and upper FOMC dates surrounding the trade date. In turn, we have derived the business day offset both above. These offsets are required to calculate the correct week periods in the regression analysis. Before we do that, it is necessary to join together the equity premium data with the FOMC cycle data.

Below we display the 100 earlier and 100 latest observations of FOMC cycle data.

cycle_data[c(1:100,(N-100):N),] %>% kable() %>% kable_styling() %>% scroll_box(height="300px", width="100%")

write_csv(cycle_data, "cycle_data.csv")

Merge the Cycle and Excess Return Data

Now we combine the FOMC cycle and equity excess return data in order to stage the regression analysis.

eq_data %>% inner_join( cycle_data, by = c("mdyDate" = "trade_date")) -> mgData

mgData %>% filter(  mdyDate > mdy("3-20-1994") ) %>% filter(mdyDate < mdy("5-07-2019")) %>%
select( mdyDate, excess_ret, lowerFOMC, offsetL, offsetU, weekNum, 
        dum_w0, dum_w2, dum_w4, dum_w6, dum_w246, dum_w0246) ->usable

dim(usable)

## [1] 6341   12

usable[c(1:100,(N-100):N),] %>% kable() %>% kable_styling() %>% scroll_box(height="300px", width="100%")

write_csv(usable, "usable.csv")

write_rds(usable, "usable.rds")

Regression Analysis of the usable data

In our regression models below, we don’t detect statistical significance in the linear regression on the coefficients of the variables. This is somewhat puzzling given that the paper is quite precise in providing data construction methodology for the FOMC cycle.

We illustrate with two regression models: one on the dummy of the even weeks and another on the offset days themselves which is more granular.

mod2 = lm(excess_ret ~ factor(dum_w0) + factor(dum_w2) + factor(dum_w4) + factor(dum_w6) , data = usable )
summary(mod2)

## 
## Call:
## lm(formula = excess_ret ~ factor(dum_w0) + factor(dum_w2) + factor(dum_w4) + 
##     factor(dum_w6), data = usable)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.187151 -0.011548  0.001456  0.013146  0.194200 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.606e-03  4.397e-04   3.652 0.000262 ***
## factor(dum_w0)1 7.927e-05  8.554e-04   0.093 0.926171    
## factor(dum_w2)1 1.619e-03  8.615e-04   1.880 0.060176 .  
## factor(dum_w4)1 7.626e-04  8.920e-04   0.855 0.392628    
## factor(dum_w6)1 1.976e-03  1.323e-03   1.493 0.135510    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02384 on 6336 degrees of freedom
## Multiple R-squared:  0.0008542,  Adjusted R-squared:  0.0002235 
## F-statistic: 1.354 on 4 and 6336 DF,  p-value: 0.2472

plot(mod2)

mod4 = lm(excess_ret ~ offsetL, data=usable)
summary(mod4)

## 
## Call:
## lm(formula = excess_ret ~ offsetL, data = usable)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.185930 -0.011545  0.001475  0.013102  0.193649 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 1.340e-03  5.634e-04   2.379   0.0174 *
## offsetL     5.103e-05  3.161e-05   1.614   0.1065  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02384 on 6339 degrees of freedom
## Multiple R-squared:  0.000411,   Adjusted R-squared:  0.0002533 
## F-statistic: 2.606 on 1 and 6339 DF,  p-value: 0.1065

plot(mod4)

Neither model appears to be statistically significant. Other variations produced the same results.

Visualizations

The 5-day mean excess return conditional on the offset days does show the pattern that we observe in the author’s paper.

usable %>% group_by( offsetL) %>% summarize( fret = mean( excess_ret)) -> fret_by_offsetL

ggplot( data = fret_by_offsetL) + geom_line(aes(x=offsetL, y= fret)) + ggtitle("5-day excess returns vs. FOMC cycle date", subtitle="1994-2019")

ggplot(data = usable, aes(x=offsetL, y=excess_ret)) + geom_boxplot()

Conclusion

We conclude and replicate the key pattern observed by the authors using the Russell 1000 Equity Index instead of the SP500 Index. During the even-numbered weeks of the FOMC cycle, the equity returns are significantly higher than during odd-numbered weeks of the cycle.

However, the regression of excess returns to dummies of the even weeks does not produce the expected results. While I suspect that implementation issues may be causing the lack of results, this is a puzzle for two reasons. The chart of excess returns vs. FOMC cycle day exactly matches the papers in the figure 1 of the authors’ paper. Secondly, the excess returns displays in a scatter plot show a high degree of dispersion within each offset day. However, there are noticeable but subtle shifts in mean on even numbered days: 0, 9, 20, 30. This suggests that the pattern may be economically significant but harder to show statistical significance.

References

• Cieslak, Anna and Morse, Adair and Annette Vissing-Jorgensen, “Stock Returns Over the FOMC Cycle”, Journal of Finance, forthcoming, [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2687614]