Momentum

Using CRSP stock data, define the universe of monthly returns that can be used in calculating momentum portfolios, as well as their ranking return, following the procedure in Daniel and Moskowitz (2016). Your output should be from 1927-2017.

Part 1

Using CRSP stock data, define the universe of monthly returns that can be used in calculating momentum portfolios, as well as their ranking return, following the procedure in Daniel and Moskowitz (2016). Your output should be from 1927-2017.

We are going to replicate the momentum data following the instructions on Kent Daniel’s website.

Steps:

Universe of stocks: Following Daniel-Moskowitz and Ken French’s procedure, I restrict the sample to common shares (share codes 10 and 11) and to securities traded in the New York Stock Echange, American Stock Exchange, or the Nasdaq Stock Exchange (exchange codes 1, 2, and 3).
Missing returns: The sample missing of both RET and DLRET are removed from dataset. And NA are labeled for their respective missing code.
Duplicates removal: there are several duplicated data in CRSP, we shall remove them before any calculation.
Delisting return calculation: We use cum-dividend returns for all the calculation. Delisting return is used when there is no listing return available. When both are avaiable, we use compounded rate of return.
Market Capitalization calculation: Since the price in the dataset is at the end of the period, we shall use 1-period lag to calculate the market capitalization. The market capitalization is computed as the product of valid share price and outstanding shares from previous month.
Sample period: Trying to include as much as possible, we can get dataset from Jan 1926 to Dec 2017.
Ranking return: to calculate the ranking return, we select the most recent month as the formation period. Only when there are no less than eight monthly returns over the past 11 months before the formation period, we take the cumulative log return as the ranking return. Also, it requires a valid share price 13 months before the formation period and valid market capitalization 1 month before the formation period.

Input Dataset

The replication starts from the input data:

Variable Name	Variable Type
PERMNO	integer
date	Date
SHRCD	integer
EXCHCD	integer
RET	character
DLRET	character
PRC	numeric
SHROUT	integer

R script

PS3_Q1 <- function(CRSP_Stocks){
  # taking universe with NYSE, Amex and Nasdaq exchanged stocks
  # only using returns of common shares
  universe <- CRSP_Stocks[(EXCHCD %in% c(1, 2, 3)) & (SHRCD %in% c(10, 11)),]
  
  # filter out the duplicates in the sample
  setkey(universe, NULL)
  unique_universe <- unique(universe)
  
  # using valid shares and price
  shrs_prc_universe <- unique_universe[, ADJPRC := abs(PRC)]
  ## pre-filter out the invalid prc and shrs ?? (let's check this later by push this down)
  shrs_prc_universe <- shrs_prc_universe[(ADJPRC > 0) & (SHROUT > 0), ]
  # extract year and month from date
  shrs_prc_universe[, c("Year", "Month") := list(year(date), month(date))]
  # calculate market capitalization in millions
  shrs_prc_universe[, mkc := ADJPRC * SHROUT / 1000]
  # get 1-month lag market capitalization
  shrs_prc_universe[, YM := as.yearmon(date)]
  shrs_prc_universe[, shifted_YM_1 := shift(YM), by = .(PERMNO)]
  shrs_prc_universe[, lag_Mkt_Cap_flag := ifelse(YM-1/12 == shifted_YM_1, 1, 0)]
  shrs_prc_universe[, lag_Mkt_Cap := ifelse(lag_Mkt_Cap_flag == 1, shift(mkc), NA)]
  
  # get t-13 month price
  shrs_prc_universe[, shifted_YM_13 := shift(YM, 13), by = .(PERMNO)]
  shrs_prc_universe[, lag_adjprc_flag := ifelse(YM-13/12 == shifted_YM_13, 1, 0)]
  shrs_prc_universe[, lag_adjprc := ifelse(lag_adjprc_flag == 1, shift(ADJPRC, 13), NA)]
  
  # calculate combined return using holding period return and delisting return
  ret_universe <- shrs_prc_universe[, .(PERMNO, EXCHCD, RET, DLRET, Year, Month, YM, lag_Mkt_Cap, lag_adjprc)]
  ret_universe[, c("RET", "DLRET") := list(as.numeric(RET), as.numeric(DLRET))]
  ret_universe[, RET := ifelse(is.na(RET), -99, RET)]
  ret_universe[, DLRET := ifelse(is.na(DLRET), -99, DLRET)]
  ret_universe[, adjret := ifelse(RET != -99, ifelse(DLRET != -99, (1+RET)*(1+DLRET)-1, RET), 
                                  ifelse(DLRET != -99, DLRET, NA))]
  
  formation_universe <- ret_universe[, .(PERMNO, EXCHCD, Year, Month, YM, lag_Mkt_Cap, adjret, lag_adjprc)]
  #check how many valid months in previous 12 months
  for(i in 1:12){
    # get shifted month and return
    formation_universe[, paste0("shifted_YM_", i) := shift(YM, i), by = .(PERMNO)]
    formation_universe[, paste0("shifted_ret_", i) := shift(adjret, i), by = .(PERMNO)]
    
    # check if there is return in previous month
    formation_universe[, paste0("shifted_flag_", i) := ifelse((YM-i/12 == eval(parse(text=paste0("shifted_YM_", i)))) 
                                                              & !is.na(eval(parse(text=paste0("shifted_ret_", i)))), 1, 0)]
    
    # remove the columns
    formation_universe[, paste0("shifted_YM_", i) := NULL]
    formation_universe[, paste0("shifted_ret_", i) := NULL]
  }
  
  # count the number of valid flag over month t-12 to month t-2
  formation_universe[, formation_flag := Reduce(`+`, .SD), .SDcols = paste0("shifted_flag_", 2:12)]
  
  # check if valid to include into portfolio
  ## At least 8 months of return data between t−12 and t−2
  ## return on month t-12 and month t-2 exist
  formation_universe[, if_include := ifelse((formation_flag >= 8) & 
                                              (shifted_flag_2 == 1) & 
                                              (shifted_flag_12 == 1) & 
                                              (!is.na(lag_Mkt_Cap)) &
                                              (!is.na(lag_adjprc)),
                                            1, 0)]
  
  # calculate ranking return
  ## use prod(x) and get log of the cumulative return
  formation_universe[, plus_ret := adjret+1]
  formation_universe[, Ranking_Ret := ifelse(if_include, log(Reduce(`*` , shift(plus_ret, 2:12))), NA)]
  
  CRSP_Stocks_Momentum <- na.omit(formation_universe[, .(Year, Month, PERMNO, EXCHCD, lag_Mkt_Cap, Ret=adjret, Ranking_Ret)])
  return(CRSP_Stocks_Momentum)
}

Output Dataset

And it gives the following output:

Variable.Name	Variable.Type	Variable.Description
Year	numeric	Year
Month	numeric	Month
PERMNO	integer	PERMNO ID issued by CRSP
EXCHCD	integer	Exchange code
lag_Mkt_Cap	numeric	Firm’s market value the previous month (in millions)
Ret	numeric	Firm’s returns
Ranking_Ret	numeric	Firm’s ranking returns

Part 2

Define the monthly momentum portfolio decile of each stock as defined by both Daniel and Moskowitz (2016) and Kenneth R. French. Your output should be from 1927-2017.

We are going to determine the breakpoints for decile portfolio following Daniel and Moskowitz’ approach and Kenneth R. French’s approach.

Steps

First we compute the breakpoints at each formation period so that each portfolio contains same number of stocks. To calculate the breakpoints following Daniel and Moskowitz’ approach, we use all firms to create quantiles. For all the formation periods each quantile contains same number of stocks.
For Kennetch French’s approach, we create breakpoints based on stocks exhcanged in NYSE only. However, the NYSE breakpoints may not include the all firm return ranges. To deal with this problem, we need to remap the head and tail quantiles of NYSE breakpoints to quantiles of all firm breakpoints.

Input Dataset

Variable.Name	Variable.Type	Variable.Description
Year	numeric	Year
Month	numeric	Month
PERMNO	integer	PERMNO ID issued by CRSP
EXCHCD	integer	Exchange code
lag_Mkt_Cap	numeric	Firm’s market value the previous month (in millions)
Ret	numeric	Firm’s returns
Ranking_Ret	numeric	Firm’s ranking returns

R script

PS3_Q2 <- function(CRSP_Stocks_Momentum){
  # DM decile quantile calculator
  DM_GetPortNums <- function(x) {
    as.integer(cut(x,
                   quantile(x, probs=0:10/10, na.rm = TRUE),
                   include.lowest=T))
  }
  
  CRSP_Stocks_Momentum_decile <- copy(CRSP_Stocks_Momentum)
  # divide by DM decile for equal stocks for all exchange code
  CRSP_Stocks_Momentum_decile[, DM_decile := DM_GetPortNums(Ranking_Ret), by = .(Year, Month)]
  
  # divide by KRF decile for equal stock for exchange code == 1 (NYSE)
  KRF_GetPortNums <- function(x){
    min1 <- min(x[EXCHCD==1, Ranking_Ret])
    max1 <- max(x[EXCHCD==1, Ranking_Ret])
    min <- min(x$Ranking_Ret)
    max <- max(x$Ranking_Ret)
    
    if(min1 >= min){
      if(max1 <= max){
        fitted_quantile <- c(min-1, quantile(x[EXCHCD==1, Ranking_Ret], probs=0:10/10, na.rm = TRUE)[2:10], max+1)
      }else{
        fitted_quantile <- c(min-1, quantile(x[EXCHCD==1, Ranking_Ret], probs=0:10/10, na.rm = TRUE)[2:11])
      }
    }else{
      if(max1 <= max){
        fitted_quantile <- c(quantile(x[EXCHCD==1, Ranking_Ret], probs=0:10/10, na.rm = TRUE)[1:10], max+1)
      }else{
        fitted_quantile <- quantile(x[EXCHCD==1, Ranking_Ret], probs=0:10/10, na.rm = TRUE)
      }
    }
    as.integer(cut(x$Ranking_Ret, 
                   fitted_quantile, 
                   include.lowest = T)) ## !! have to update the quantile for each YM
    ## need modification so that the top and bottom quantile is inf
    ## sol: first find min and max of EXCHCD==1 and global min and max
  }
  CRSP_Stocks_Momentum_decile[, KRF_decile := KRF_GetPortNums(.SD), by = .(Year, Month)]
  CRSP_Stocks_Momentum_decile[, Ranking_Ret := NULL]
  CRSP_Stocks_Momentum_decile[, EXCHCD := NULL]
  return(CRSP_Stocks_Momentum_decile)
}

Output Dataset

Variable.Name	Variable.Type	Variable.Description
Year	numeric	Year
Month	numeric	Month
PERMNO	integer	PERMNO ID issued by CRSP
lag_Mkt_Cap	numeric	Firm’s market value the previous month (in millions)
Ret	numeric	Firm’s returns
DM_decile	integer	Firm’s momentum decile as defined by Daniel & Moskowitz
KRF_decile	integer	Firm’s momentum decile as defined by Kenneth R. French

Part 3

Calculate the monthly momentum portfolio decile returns as defined by both Daniel and Moskowitz (2016) and Kenneth R. French. Your output should be from 1927-2017.

From previous question, we have times series of deciles. I compute the decile portfolio returns as follows:

Sample period: Monthly from Jan 1927 to December 2017.
Value-weighted portfolio: A decile portfolio weighted by total market capitalization and rebalanced monthly.
Definition of portfolio weights: For value-weighted portfolio, the weight is computed from the weight of lagged market capitalization of respective stock. All the decile portolio return is calculated as value-weighted return.

Input Dataset

Variable.Name	Variable.Type	Variable.Description
Year	numeric	Year
Month	numeric	Month
PERMNO	integer	PERMNO ID issued by CRSP
EXCHCD	integer	Exchange code
lag_Mkt_Cap	numeric	Firm’s market value the previous month (in millions)
Ret	numeric	Firm’s returns
Ranking_Ret	numeric	Firm’s ranking returns

And

Variable.Name	Variable.Type
Year	numeric
Month	numeric
Market_minus_Rf	numeric
SMB	numeric
HML	numeric
Rf	numeric

R script

PS3_Q3 <- function(CRSP_Stocks_Momentum_decile, FF_mkt){
  # calculate value-weighted return in each decile portfolio
  DM_Momentum_returns <- CRSP_Stocks_Momentum_decile[, .(DM_Ret = sum(Ret*lag_Mkt_Cap / sum(lag_Mkt_Cap)), decile = DM_decile), by = .(DM_decile, Year, Month)]
  KRF_Momentum_returns <- CRSP_Stocks_Momentum_decile[, .(KRF_Ret = sum(Ret*lag_Mkt_Cap / sum(lag_Mkt_Cap)), decile = KRF_decile), by = .(KRF_decile, Year, Month)]
  # retrieve decile portfolio return columns
  DM_Momentum <- DM_Momentum_returns[, .(Year, Month, decile, DM_Ret)]
  KRF_Momentum <- KRF_Momentum_returns[, .(Year, Month, decile, KRF_Ret)]
  # merge with risk free rate
  setkey(DM_Momentum, Year, Month, decile)
  setkey(KRF_Momentum, Year, Month, decile)
  aa <- DM_Momentum[KRF_Momentum,]
  rf <- FF_mkt[, .(Year, Month, Rf)]
  setkey(rf, Year, Month)
  setkey(aa, Year, Month)
  # fast join
  CRSP_Stocks_Momentum_returns <- rf[aa, ]
  # reorder the columns
  setcolorder(CRSP_Stocks_Momentum_returns, c("Year", "Month", "decile", "DM_Ret", "KRF_Ret", "Rf"))
  return(CRSP_Stocks_Momentum_returns)
}

Output Dataset

Variable.Name	Variable.Type	Variable.Description
Year	numeric	Year
Month	numeric	Month
decile	integer	Momentum decile
DM_Ret	numeric	Decile (as defined by Daniel & Moskowitz (2016)) return
KRF_Ret	numeric	Decile (as defined by Kenneth R. French) return
Rf	numeric	Riskless rate

For illustration purpose, we plot log cumulative returns of the KRF momentum decile portfolio and DM momentum decile portfolios.

Momentum

Part 4

Replicate Table 1 in Daniel and Moskowitz (2016), except for $\alpha$, $t(\alpha)$, $\beta$, and $sk(d)$ rows, and the Market column. Match the format and methodology to the extent possible.

From previous question, we have a times series of decile portfolio returns for each month. I compute the required statistics as follows:

Sample period: Monthly from Jan 1927 to December 2017.
Excess return: Annualized excess return is calcualted as mean monthly excess return times 12.
Volatility: Volatility is calculated as standard deviation of excess return times square root of 12.
Annualized Sharpe Ratio: For annualized Sharpe ratio, we divide the excess return by volatility.
Skewness: for all portfolios except WML, the we calculate the full-period realized skewness of monthly log returns of the decile portfolio. For WML return, we calculate the skewness as $log(1 + r_{WML}+r_f)$.
WML portfolio: the WML portfolio stands for the winner-minus-loser portfolio which is long the Decile 10 portfolio and short the Decile 1 portfolio.

Output Dataset

The output is a $4 \times 11$ numeric matrix, reproducing part of Table 1 in Daniel & Moskowitz (2016). It matches the format and methodology to the extent possible.

Return statistics	1	2	3	4	5	6	7	8	9	10	WML
Excess_Ret	-2.623	2.530	3.065	6.378	7.331	7.054	9.082	10.283	11.47	15.37	17.99
Vol	36.67	30.38	26.05	23.14	21.57	20.38	19.56	19.15	20.52	23.94	29.98
SR	-0.07152	0.08328	0.1176	0.2755	0.3398	0.3460	0.4642	0.5367	0.5590	0.64207	0.6002
skm	0.07515	-0.1224	-0.1662	0.1291	-0.08925	-0.2373	-0.6442	-0.5490	-0.7583	-0.8084	-5.106

Part 5

Calculate the correlation of your portfolio returns with the Daniel and Moskowitz (2016) breakpoints (by decile), to the portfolio returns on Daniel’s website. Also calculate the correlation of your portfolio returns with the Kenneth R. French breakpoints (by decile), to the portfolio returns on French’s website. Round to 4 decimal places. Correlations should be calculated from 1927-2017.

Steps

Data source: Kent Daniel provides data download on his website. We can also download the NYSE breakpoints decile portfolio from Kenneth French’s website.
Sample range: since Daniel and Moskowitz only provide data from Jan 1927 to Dec 2016, we shall compute the correlation with the maximum available time horizon.
Correlation: For each decile portfolio, we calculate the full-period correaltion between the monthly return of our replicated data and the data given by Kenneth French and Daniel and Moskowitz.

Output Dataset

The output is a 2 by 11 numeric matrix, with the correlations between your estimated DM momentum portfolio returns and the DM momentum portfolio returns on Daniel’s website, and the correlations between your estimated KRF momentum portfolio returns and the KRF momentum portfolio returns on French’s website.

variable	1	2	3	4	5	6	7	8	9	10	WML
DM correlation	0.9973	0.9972	0.9978	0.9977	0.9978	0.9981	0.9986	0.9989	0.9984	0.9981	0.9948
KRF correlation	0.9977	0.9984	0.9977	0.9977	0.9976	0.9968	0.9978	0.9986	0.9983	0.9988	0.9955

The difference between the replicated momentum returns and the outcome from Daniel and Moskowitz (2016) and Kenneth French is not zero because of different ways in data processing and sample selection. Part of the deviation comes from the use of data source. Notice that the correlations at higher deciles have closer values to 1, which indicates the breakpoints choice may differ in the lower deciles.

Part 6

Has the momentum anomaly worked in the past few years? Show some empirical evidence.

To examine the performance of momentum strategy in the recent years, we shall analyse the cumulative gains from investments of most recent 10 years. The data is from the output of part 3.

Performance

As shown in the plot, compared to market cumulative return, the decile 1 and decile 10 return have no superior performance. And we do observe several severe drawdown following the 2008-2009 financial crisis and 2015-2016 crash down. As for the performance of WML portfolio, it has relatively poor performance over this time period. This mostly comes from the short side (the past losers) as it rebounces much more than long side after crash. Overall, the momentum return has the worst performance in the above asset class over recent years.