JEF Paper Coding (2022) - v3.3 Terciles (For Monthly Data

Data cleaning and Preparation

Information about initial coding setup:

freq sets the data frequency for the following analysis, 12 for monthly data, 4 for quarterly data and 1 for annual data.
start.ym gives the earliest reasonable starting point of the series, which is January 1966, based on the available number of firms in the data set.
After the preliminary data cleaning, port_market is the market portfolio data (including NYSE, NASDAQ and AMEX), ports_all contains different deciles. All the data are stored in the file named as market.names and data.names. I’ve finished creating the measures based on characteristic deciles, so I’ll have a close look at your results shortly. The decile data is attached. As mentioned, these are based on a single characteristic sort, which will hopefully provide new insight into characteristic based predictability. The characteristics are as follows:

Size (size)
Book-to-market ratio (bm)
Operating profitability (op)
Investment (inv); and
Momentum (mom)

The filenames indicate which is which, and the deciles are denoted as D1 (low value on the characteristic) to D10 (high value on the characteristic), consistent with both Maio (2014) and Maio (2016), with the breakpoints identified according to the NYSE distribution. The basis of the sorts are consistent with Fama and French (2015) (and their earlier work) in that re-sorting occurs only one a year. An alternative is to re-sort monthly, which I’ve done for other papers applying a similar approach, but doing so comes with its own drawbacks.

RF denotes the risk-free rate, which is the average of the bid and ask.

# 0. record datasets ----
## 0.1 initial value setup ----
freq = 12 # the frequency of the data <- 12 for monthly; 4 for quarterly; 1 for annually
start.ym = as.yearmon(1966) # the starting time

month_select <- function(freq = freq) { # return the months for differnt time frequency
  if (freq == 12) {
    return(c("January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November","December"))
  }
  if (freq == 4) {
    return(c("March", "June", "September", "December"))
  }
  if (freq == 1) {
    return("December")
  }
}
freq_name <- function(freq = freq) {
  if (freq == 12) {
    return("monthly")
  }
  if (freq == 4) {
    return("quarterly")
  }
  if (freq == 1) {
    return("annual")
  }
}

## 0.2 preliminary data cleaning ----
port_market <- read.csv("Allfirms_comp2.csv") %>%
  as.tbl() %>%
  # mutate(E = rollmeanr(E, k = 12, fill = NA_real_)) %>%
  mutate(month = as.yearmon(as.Date(date)),
         vwret = rollsumr(vwret, k = 12/freq, fill = NA)) %>%
  filter(months(month) %in% month_select(freq = freq))
write.csv(x = port_market, file = "market_Allfirms.csv")

ports_all <- read.csv("terciles_size.csv") %>%
  as.tbl() %>%
  mutate(month = as.yearmon(as.Date(jdate)), # (, format = "%d/%m/%Y")
         port = as.character(port))
ports <- unique(ports_all$port) # identifiers for portfolios
for (p in ports) {
  ports.dt <- ports_all %>%
    filter(port == p) %>%
    arrange(month) %>%
    # mutate(vwret = rollsumr(vwret, k = 12/freq, fill = NA)) %>%
    filter(months(month) %in% month_select(freq = freq)) # %>%
    # mutate(E = rollmeanr(E, k = 12, fill = NA_real_))
  ports.dt <- ports.dt[, -1]
  write.csv(x = ports.dt, file = paste("port_", p, ".csv", sep = ""))
}

## 0.3 name portfolios and predictors ----
market.names <- list.files(pattern = "market_")
data.names <- list.files(pattern = "port_") # paste("port_", "D", 1:10, ".csv", sep = "") # data for portfolios
id.names <- c("Market", "B", "M", "S") # c("Market", paste("D", 1:10, sep = "")) # set plot names # c("Market", "B", "M", "S") # c("Market", "H", "M", "L") # c("Market", "W", "N", "L")
ratio_names <- c("DP", "PE", "EY", "DY", "Payout") # potential predictors

## 0.4 risk-free rate
RF <- read.csv("Rfree_t30.csv") %>% # record the risk free rate
  as.tbl() %>% # as the average of the bid and ask.
  select(-X) %>%
  mutate(month = as.yearmon(month)) %>%
  filter(months(month) %in% month_select(freq = freq)) %>%
  filter(month >= start.ym)

Notes: Seems that the big-value and small-growth portfolios include less firms comparing the other four characteristic portfolios, around half of them.

Figure 1 - Log Cumulative Index

Log cumulative realised portfolio return components for seven portfolios - the market portfolio and six size and book-to-market equity ratio sorted portfolios. All following figures decmonstrate the monthly realised price-earnings ratio growth (gm), earnings growth (ge), dividend-price (dp) and the portfolio return index (r) with the values in January 1966 as zero for all portfolios.

# TABLE-1. summary statistics ----
TABLE1.uni <- list() # the univariate statistics
TABLE1.cor <- list() # the correlation matrixs

PE.df <- data.frame(month = port_market$month[port_market$month >= start.ym])
EY.df <- data.frame(month = port_market$month[port_market$month >= start.ym])
DP.df <- data.frame(month = port_market$month[port_market$month >= start.ym])

## (1*) summary tables for Summary & Correlations ----
c <- 0
for (id in c(market.names, data.names)) {
  c <- c + 1
  # print(id); print(id.names[c])
  
  ## 1. read the data ----
  data_nyse <- read.csv(id) %>%
    as.tbl() %>%
    mutate(month = as.yearmon(month)) %>%
    filter(month >= start.ym) %>% # start from "Jan 1966"
    select(month, r = vwret, P, E, D) %>%
    mutate(DP = D / P, # these are adjusted by the log transformation
           PE = P / E,
           EP = E / P,
           EY = E / lag(P), # earnings yield
           DY = D / lag(P), # dividend yield
           Payout = D / E) # payout ratios
  
  PE.df <- cbind.data.frame(PE.df, data_nyse$PE)
  EY.df <- cbind.data.frame(EY.df, data_nyse$EY)
  DP.df <- cbind.data.frame(DP.df, data_nyse$DP)
  
  ## 2. return decomposition ----
  data_decompose <- data_nyse %>%
    mutate(r = r, # cts returns = log total returns
           gm = log(PE) - lag(log(PE)), # multiple expansion rate
           ge = log(E) - lag(log(E)), # earnings growth rate
           dp = log(1 + DP/freq)) %>% # only 1/12 of the dividends
    na.omit()
  
  ## 3. summary-Stat ----
  ar1.coef <- function(x) {
    return(as.numeric(lm(x ~ lag(x))$coefficients[2]))
  } # return the function value of the coefficient for the AR(1) model
  
  comp_summary.dt <- data_decompose %>%
    select(gm, ge, dp, r) %>%
    describe() %>%
    mutate(mean = mean * 100,
           sd = sd * 100,
           median = median * 100,
           min = min * 100,
           max = max * 100) %>%
    select(Mean = mean, Median = median, SD = sd, Min = min, Max = max, Skew = skew, Kurt = kurtosis) %>%
    round(digits = 4)
  
  comp_summary.dt$"AR(1)" <- data_decompose %>%
    select(gm, ge, dp, r) %>%
    apply(2, ar1.coef) %>%
    round(digits = 4)
  
  ### Store the summary stat
  # print(paste("Data starts from ", first(data_decompose$month), " and ends in ", last(data_decompose$month), ".", sep = ""))
  TABLE1.uni[[id.names[c]]] <- comp_summary.dt
  
  ## 4. correlations ----
  comp_cor <- data_decompose %>% select(gm, ge, dp, r) %>% cor()
  TABLE1.cor[[id.names[c]]] <- comp_cor
  
  # Figure-1. cumulative realised return components ---- 
  # jpeg(filename = paste("Figure1_", id.names[c], ".jpeg", sep = ""), width = 550, height = 350)
  par(mar = c(2, 4, 2, 1))
  cum_components.ts <- data_decompose %>%
    select(r, gm, ge, dp) %>%
    apply(2, cumsum) %>%
    ts(start = data_decompose$month[1], frequency = freq)
  plot.ts(cum_components.ts, plot.type = "single", lty = 1:4, main = id.names[c], cex.main = 1, 
          xlab = NULL, ylab = "Cumulative Return and Components Indices")
  legend("topleft",
         legend = c("Total return", "Price earnings growth",
                    "Earnings growth", "Dividend price"),
         lty = 1:4,
         cex = 1.0) # text size
  # dev.off()
  par(mar = c(5, 4, 4, 2) + 0.1)
}

write.csv(TABLE1.uni, file = "table_1.uni.csv")
write.csv(TABLE1.cor, file = "table_1.cor.csv")

.

Table 1 - Summary statistics of returns components

The correlations between gm and ge might be a bit too high comparing to Ferreira and Santa-Clara (2011). Need to check the code again.

Need to go back to the construction process of Prof Robert Shiller’s CAPE.

‘kable’ for Table Creation

Table 1 - Summary statistics of returns components
monthly data starts from Feb 1966 and ends in Dec 2019.
	Panel A: univariate statistics								Panel B: Correlations
	Mean	Median	SD	Min	Max	Skew	Kurt	AR(1)	gm	ge	dp	r
Market
gm	0.02	-0.03	3.12	-15.26	13.28	-0.19	4.42	0.92	1.00	-0.51	-0.03	0.07
ge	0.76	1.11	5.34	-22.01	19.34	-0.50	2.44	0.33	-0.51	1.00	-0.03	0.81
dp	0.28	0.27	0.09	0.09	0.50	0.14	-0.70	0.98	-0.03	-0.03	1.00	-0.03
r	0.94	1.25	4.43	-22.48	16.58	-0.51	1.85	0.05	0.07	0.81	-0.03	1.00
B
gm	0.15	-0.03	11.91	-119.12	111.78	0.26	35.03	0.08	1.00	-0.95	-0.05	0.07
ge	0.41	0.89	12.18	-107.82	108.30	-0.34	25.30	0.09	-0.95	1.00	0.03	0.24
dp	0.29	0.28	0.13	0.08	0.64	0.50	-0.82	0.99	-0.05	0.03	1.00	-0.02
r	0.87	1.02	4.23	-18.36	17.26	-0.34	1.52	0.03	0.07	0.24	-0.02	1.00
M
gm	-0.33	-0.07	13.42	-150.27	103.61	-1.86	40.00	0.10	1.00	-0.94	-0.06	0.04
ge	1.20	1.49	13.98	-97.77	150.15	1.61	30.88	0.10	-0.94	1.00	0.03	0.29
dp	0.33	0.27	0.20	0.09	1.50	1.88	4.54	0.87	-0.06	0.03	1.00	-0.06
r	1.09	1.36	5.23	-27.29	24.97	-0.48	2.44	0.09	0.04	0.29	-0.06	1.00
S
gm	-0.15	-0.07	13.13	-111.23	106.71	-0.38	28.22	0.09	1.00	-0.91	-0.02	0.06
ge	1.23	1.89	13.92	-96.48	109.65	0.43	17.75	0.12	-0.91	1.00	0.01	0.35
dp	0.65	0.33	0.80	0.11	4.78	2.40	5.33	1.00	-0.02	0.01	1.00	-0.05
r	1.11	1.51	6.28	-30.68	30.86	-0.31	2.43	0.15	0.06	0.35	-0.05	1.00
Note: Panel A in this table presents mean, median, standard deviation (SD), minimum, maximum, skewness (Skew), kurtosis (kurt) and first-order autocorrelation coefficient of the realised components of stock market returns and six size and book-to-market equity ratio sorted portfolios. These univariate statistics for each portfolios are presented separately. gm is the continuously compounded growth rate in the price-earnings ratio. ge is the continuously compounded growth rate in earnings. dp is the log of one plus the dividend-price ratio. r is the portfolio returns. Panel B in this table reports correlation matrices for all seven portfolios. The sample period starts from Feburary 1966 and ends in December 2019.

Figure 3 - Cumulative OOS R-sqaure Difference and Cumulative SSE Difference

The cumulative OOS R-square figures show the out-of-sample cumulative R-square up to each month from predictive regressions with listed predictors and from the sum-of-the-parts (SOP) method for each portfolio. The cumulative SSE difference plots indicates the out-of-sample performance of each model. These are evaluated by the cumulative squared prediction errors of the NULL minus the cumulative squared predictirion error of the ALTERNATIVE. The NULL model is the historical mean model, while the ALTERNATIVE model is either the predictive regression model or the SOP model. An incresae in the line suggests better performance of the ALTERNATIVE model and a decrease suggests that the NULL model is better.

Several points to note in the coding:

The dividend-price ratio (‘DP’ hereafter) is calculated as the log of 1 plus the frequency-adjusted dividend to price ratio, rather than using the annual dividend. As by this return decomposition, the expected amount of dividend payout in each period should be adjusted by the frequency of the data in the analysis. \[ dp_t = \log (1 + \frac{\tilde{D}_t}{P_t}) = \log (1 + \frac{D_t / n}{P_t}) \text{,} \] where $D_t$ is the annual dividend payment and $n$ is the data frequency (e.g. $n = 1$ for annual data and $n = 12$ for monthly data) and $\tilde{D}_t$ is the freqency-adjusted dividend payment for period $t$.
The SOP method by Ferreira and Santa-Clara (2011) decomposes the portfolio return into three components, namely the earnings growth, the prie multiple expansion and the next period dividend-price ratio. Here to generate the SOP prediction, we use the rolling mean of past earnings growth as the expected growth of the next period (denoted as ge1). However, there are other choices, such as recursive means in ge2 and ge3.
critica.value = TRUE is the option whether to use boostrap method to calculate the MSE-F critical values. This is used in function Boot_MSE.F.
The authors should evaluate the significance of the MSE−F statistic by using the theoret- ical distribution derived in McCracken (2007). The bootstrap-based inference (presented in Pages 9-10) can represent a robustness check and moved to an appendix. Further- more, the authors can also include in the main results the related out-of-sample statistic proposed by Clark and West (2007), which follows a standard Normal distribution. Therefore, readjust the Boot_MSE.F function.
Column McCracken in Table 2 (line 604) gives the significance of the out-of-sample $MSE–F$ statistic of McCracken (2007). $***$, $**$, and $*$ denote significance at the 1%, 5%, and 10% level, respectively. Please refer to the Table 4 on P749 in McCracken (2007) with $k_2 = 1$ and $\pi = P/R = \frac{\text{Number of out-of-sample forecasts}}{\text{Number of observations used to form the first forecast}} = 1.6$.

Examples are in P742-743. See detailed notations

## [1] "market_Allfirms.csv"
## [1] "Market"
## ##------ Tue May 17 21:06:04 2022 ------##

## Note: Using an external vector in selections is ambiguous.
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## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.

## Note: Using an external vector in selections is ambiguous.
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## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.

## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(uncond)` instead of `uncond` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
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## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
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## [1] "OOS R Squared: 0.0086"
## [1] "MSE-F: 2.9095"

## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(predictor)` instead of `predictor` to silence this message.
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## [1] "IS R Squared: 0.0106"
## [1] "OOS R Squared: -0.0156"
## [1] "MSE-F: -5.1496"
## [1] "IS R Squared: 0.0045"
## [1] "OOS R Squared: -0.002"
## [1] "MSE-F: -0.6624"
## [1] "IS R Squared: 0.0051"
## [1] "OOS R Squared: -0.0018"
## [1] "MSE-F: -0.6056"
## [1] "IS R Squared: 0.012"
## [1] "OOS R Squared: -0.017"
## [1] "MSE-F: -5.5903"
## [1] "IS R Squared: 1e-04"
## [1] "OOS R Squared: -0.005"
## [1] "MSE-F: -1.6534"

## Warning in window.default(x, ...): 'start' value not changed

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## [1] "port_B.csv"
## [1] "B"
## ##------ Tue May 17 21:06:08 2022 ------##
## [1] "OOS R Squared: -0.0091"
## [1] "MSE-F: -3.0294"

## [1] "IS R Squared: 0.006"
## [1] "OOS R Squared: -0.0258"
## [1] "MSE-F: -8.4119"
## [1] "IS R Squared: 0.0064"
## [1] "OOS R Squared: -0.0102"
## [1] "MSE-F: -3.3874"
## [1] "IS R Squared: 0.0064"
## [1] "OOS R Squared: -0.0099"
## [1] "MSE-F: -3.29"
## [1] "IS R Squared: 0.0062"
## [1] "OOS R Squared: -0.0249"
## [1] "MSE-F: -8.1436"
## [1] "IS R Squared: 0.0052"
## [1] "OOS R Squared: -0.0019"
## [1] "MSE-F: -0.6398"

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## [1] "port_M.csv"
## [1] "M"
## ##------ Tue May 17 21:06:10 2022 ------##
## [1] "OOS R Squared: 0.01"
## [1] "MSE-F: 3.4986"

## [1] "IS R Squared: 0.0029"
## [1] "OOS R Squared: -6e-04"
## [1] "MSE-F: -0.2029"
## [1] "IS R Squared: 0.0061"
## [1] "OOS R Squared: -0.0028"
## [1] "MSE-F: -0.9778"
## [1] "IS R Squared: 0.0065"
## [1] "OOS R Squared: -0.0036"
## [1] "MSE-F: -1.2333"
## [1] "IS R Squared: 0.0036"
## [1] "OOS R Squared: -8e-04"
## [1] "MSE-F: -0.2653"
## [1] "IS R Squared: 0.0043"
## [1] "OOS R Squared: -0.0058"
## [1] "MSE-F: -1.9877"

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## [1] "port_S.csv"
## [1] "S"
## ##------ Tue May 17 21:06:13 2022 ------##
## [1] "OOS R Squared: -0.0241"
## [1] "MSE-F: -8.1694"

## [1] "IS R Squared: 6e-04"
## [1] "OOS R Squared: -0.0087"
## [1] "MSE-F: -2.9973"
## [1] "IS R Squared: 0.0043"
## [1] "OOS R Squared: -0.0079"
## [1] "MSE-F: -2.7165"
## [1] "IS R Squared: 0.0051"
## [1] "OOS R Squared: -0.0096"
## [1] "MSE-F: -3.3034"
## [1] "IS R Squared: 0.0012"
## [1] "OOS R Squared: -0.0124"
## [1] "MSE-F: -4.2363"
## [1] "IS R Squared: 0.0018"
## [1] "OOS R Squared: -0.0093"
## [1] "MSE-F: -3.1897"

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Table 2 - Forecasts of portfolio returns

This table demonstrates the in-sample and out-of-sample R-squares for the market and six size and book-to-market equity ratio sorted portfolios from predictive regressions and the Sum-of-the-Parts method. IS R-squares are estimated using the whole sample period and the OOS R-squares are calculated compare the forecast error of the model against the historical mean model. The full sample period starts from Feb 1966 to December 2019 and the IS period is set to be 20 years with forecsats beginning in Feb 1986. The MSE-F statistics are calculated to test the hypothesis $H_0: \text{out-of-sample R-squares} = 0$ vs $H_1: \text{out-of-sample R-squares} \neq 0$.

Predictors here are all in log terms.

gt(table2.df, rowname_col = "rowname", groupname_col = "portname") %>%
  tab_header(title = "Table 2 - Forecasts of portfolio returns",
             subtitle = paste(freq_name(freq = freq), " data starts from ", first(data_decompose$month), " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_number(columns = 1:4, decimals = 6, suffixing = TRUE)

Table 2 - Forecasts of portfolio returns
monthly data starts from Feb 1966 and ends in Dec 2019.
	IS_r.squared	OOS_r.squared	MAE_A	MSE_F	McCracken
Market
DP	0.010630	−0.015612	0.032019	−5.149649
PE	0.004546	−0.001981	0.031621	−0.662374
EY	0.005117	−0.001811	0.031644	−0.605646
DY	0.011952	−0.016971	0.032091	−5.590273
Payout	0.000117	−0.004960	0.031203	−1.653375
SOP	NA	0.008585	0.030951	2.909518	**
B
DP	0.005961	−0.025757	0.031528	−8.411896
PE	0.006377	−0.010215	0.031146	−3.387433
EY	0.006438	−0.009918	0.031139	−3.290031
DY	0.006214	−0.024915	0.031511	−8.143588
Payout	0.005205	−0.001913	0.030937	−0.639774
SOP	NA	−0.009098	0.031423	−3.029422
M
DP	0.002918	−0.000587	0.038096	−0.202926
PE	0.006102	−0.002834	0.038407	−0.977757
EY	0.006531	−0.003577	0.038435	−1.233305
DY	0.003642	−0.000767	0.038156	−0.265276
Payout	0.004318	−0.005778	0.038372	−1.987703
SOP	NA	0.009982	0.038303	3.498631	**
S
DP	0.000600	−0.008738	0.043996	−2.997267
PE	0.004259	−0.007913	0.044665	−2.716472
EY	0.005128	−0.009639	0.044708	−3.303387
DY	0.001176	−0.012395	0.043988	−4.236312
Payout	0.001765	−0.009304	0.044914	−3.189662
SOP	NA	−0.024111	0.044569	−8.169418

Figure 4 - Monthly return predictions

Here I only present the monthly predictions of the historical mean model, the SOP method and the predictive regressions based on the dividend-price ratio and the earnings-price ratio.

## [1] "market_Allfirms.csv"
## [1] "Market"
## ##------ Tue May 17 21:06:18 2022 ------##

## [1] "port_B.csv"
## [1] "B"
## ##------ Tue May 17 21:06:20 2022 ------##

## [1] "port_M.csv"
## [1] "M"
## ##------ Tue May 17 21:06:24 2022 ------##

## [1] "port_S.csv"
## [1] "S"
## ##------ Tue May 17 21:06:26 2022 ------##

Figure 5 - Trading Performance (with no trading restrictions)

Markowitz optimal weight on the risky portfolio using the SOP method and the historical mean model.
Markowitz optimal weight on the risky portfolio using all models.
Out-of-sample performance of the trading strategies - the SOP method, the historical mean model and the buy-and-hold strategy - with no trading restrictions.

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## log, : 804 y values <= 0 omitted from logarithmic plot

Table 3 - Certaint equivalent gains

Trading Strategies: certaint equivalent gains

This table shows the out-of-sample portfolio choice results at monthly frequencies from predictive regressions and the SOP method. The trading strategy for each portfolio is designed by optimally allocating funds between the risk-free asset and the corresponding risky portfolio. The certainty equivalent return is $\overline{rp} - \frac{1}{2} \gamma \hat{\sigma}_{rp}^{2}$ with a risk-aversion coefficient $\gamma = 3$. The annualised certainty equivalent gain (in percentage) is the monthly certainty equivalent gain multiplied by the corresponding frequency (e.g. 12 for monthly data).

dt <- table3.df %>%
  filter(rowname %in% c(ratio_names, "sop_simple")) %>%
  select(CEGs_annualised, rowname, portname)

as.data.frame(matrix(dt$CEGs_annualised, byrow = F, nrow = length(ratio_names) + 1, ncol = length(id.names))) %>%
  `colnames<-`(unique(dt$portname)) %>%
  mutate(Variable = unique(dt$rowname)) %>%
  # round(digits = 4) %>%
  as.tbl() %>%
  select(Variable, unique(dt$portname)) %>%
  gt(rowname_col = "Variable") %>%
  tab_header(title = "Table 3 - Trading Strategies: certainty equivalent gains",
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month) + 20, " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_percent(columns = 2:(length(id.names)+1), decimals = 2)

Table 3 - Trading Strategies: certainty equivalent gains
Monthly data starts from Feb 1986 and ends in Dec 2019.
	Market	B	M	S
sop_simple	1.10%	−8.47%	−2.94%	−11.04%
DP	−2.85%	−4.62%	−8.49%	5.57%
PE	0.84%	−9.68%	−3.80%	−28.92%
EY	0.98%	−8.98%	−4.03%	−32.40%
DY	−2.36%	−4.35%	−10.14%	−1.23%
Payout	−4.13%	−5.90%	−0.66%	−53.35%

Table 4 - Sharpe ratio Gains

Trading Strategies: Sharpe ratio Gains

This table presents the Sharpe ratio of the out-of-sample performance of trading strategies, allocating funds between risk-free and risky assets for each portfolio. The annualised Sharpe ratio is generated by multipling the monthly Sharpe ratio by square root of the corresponding frequency (e.g. $\sqrt{12}$ for monthly data).

dt <- table4.df %>%
  filter(rowname %in% c(ratio_names, "sop_simple")) %>%
  select(SRG_annualised, rowname, portname)

as.data.frame(matrix(dt$SRG_annualised, byrow = F, nrow = length(ratio_names) + 1, ncol = length(id.names))) %>%
  `colnames<-`(unique(dt$portname)) %>%
  mutate(Variable = unique(dt$rowname)) %>%
  # round(digits = 4) %>%
  as.tbl() %>%
  select(Variable, unique(dt$portname)) %>%
  gt(rowname_col = "Variable") %>%
  tab_header(title = "Table 4 - Trading Strategies: Sharpe ratio gains", 
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month) + 20, " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_number(columns = 2:(length(id.names)+1), decimals = 4)

Table 4 - Trading Strategies: Sharpe ratio gains
Monthly data starts from Feb 1986 and ends in Dec 2019.
	Market	B	M	S
sop_simple	0.0524	−0.3751	−0.1623	0.0446
DP	−0.1667	−0.3395	−0.3162	0.2043
PE	0.0674	−0.3807	−0.1932	−0.2605
EY	0.0827	−0.3679	−0.1981	−0.2608
DY	−0.1441	−0.3201	−0.3509	0.1782
Payout	−0.1547	−0.2724	−0.0382	−0.3180

Figure 6 - Sensitivity of Certainty Equivalent Gains relative to Risk-Aversion level

This figure presents the out-of-sample portfolio choice results at monthly frequency from bivariate predictive regressions and the SOP method with different levels of risk-aversion. To show that our previous results hold with respect to investors with different levels of risk aversion, we evaluate the changes in certainty equivalent gains with respect to the changes in the level of risk-aversion. The results of the trading strategy reported here are without trading restrictions (as in Table 5), allocating funds between the risk-free asset and the risky equity portfolio. The portfolio choice results are evaluated in the certainty equivalent return with relative risk-aversion coefficient $\gamma$, with ${$0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$}$. Risky equity portfolios include the market portfolio and six size and book-to-market equity sorted portfolios, BH, BM, BL, SH, SM and SL. The annualised certainty equivalent gain is the monthly certainty equivalent gain multiplied by twelve. The sample period is from February 1966 to December 2019 and the out-of-sample period starts in March 1986.

## [1] "Market"
## ##------ Tue May 17 21:06:32 2022 ------##
## [1] "B"
## ##------ Tue May 17 21:06:32 2022 ------##
## [1] "M"
## ##------ Tue May 17 21:06:32 2022 ------##
## [1] "S"
## ##------ Tue May 17 21:06:32 2022 ------##

## Warning: Removed 3 rows containing missing values (geom_point).

## Warning: Removed 3 row(s) containing missing values (geom_path).

Table 5 - MSPE-adjusted Statistic

MSPE-adjusted Statistic

This table presents the MSEP-adjusted Statistics, evaluating the statistical significance of the out-of-sample R-squared statistics of each model in the corresponding portfolio.

See Rapach et al., (2010) and Clark and West (2007) for the detailed procedure.

table5.df <- data.frame()
for (port in names(TABLE5)) {
  pt <- TABLE5[[port]]
  pt$rowname <- rownames(pt)
  pt$portname <- port
  colnames(pt)[4] <- "star"
  table5.df <- rbind.data.frame(table5.df, pt)
}

table5.output <- gt(table5.df, rowname_col = "rowname", groupname_col = "portname") %>%
  fmt_percent(columns = vars(OOS_r.squared, mspe_pvalue), decimals = 2) %>%
  fmt_number(columns = vars(mspe_t), decimals = 4) %>%
  tab_header(title = "Table 5 - MSPE-adjusted Statistic",
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month), " and ends in ", last(data_decompose$month), ".", sep = ""))

table5.output

Table 5 - MSPE-adjusted Statistic
Monthly data starts from Feb 1966 and ends in Dec 2019.
	OOS_r.squared	mspe_t	mspe_pvalue	star
Market
DP	−1.04%	0.7486	22.73%
PE	−0.23%	0.6344	26.31%
EY	−0.17%	0.7180	23.66%
DY	−0.86%	0.9945	16.03%
Payout	−0.79%	−1.4743	92.94%
SOP	0.47%	1.2007	11.53%
B
DP	−1.66%	0.0614	47.55%
PE	−1.23%	0.5449	29.31%
EY	−1.14%	0.5886	27.82%
DY	−1.47%	0.1844	42.69%
Payout	−0.59%	0.6676	25.24%
SOP	−1.77%	1.2273	11.02%
M
DP	−1.44%	−0.8063	78.97%
PE	−0.97%	0.7000	24.22%
EY	−1.01%	0.7665	22.19%
DY	−1.64%	−0.5946	72.38%
Payout	−0.60%	0.9553	17.00%
SOP	−0.52%	1.0021	15.84%
S
DP	−2.27%	−0.4548	67.53%
PE	−2.06%	0.0963	46.17%
EY	−2.27%	0.2896	38.61%
DY	−3.14%	−0.0778	53.10%
Payout	−1.90%	−0.3536	63.81%
SOP	−4.64%	0.3680	35.65%

JEF Paper Coding (2022) - v3.3 Terciles (For Monthly Data - Size)

Hongyi Xu - Stockholm School of Economics

5/4/2022