In JustinMShea/ExpectedReturns: Reproduction of Investment Strategies presented in Antti Ilmanen's Expected Returns and Investing Amid Low Expected Returns
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
echo = TRUE
)
# Mentors, please feel free to add yourself to the authors' field if you wish.
Introduction
Our main reference is \textcite{ilmanen-2011} in that we want to reproduce selected key results the author discusses on factor-based investing. However, to achieve those results many more studies from the academic and practitioners' literature need to be implemented first. An excellent resource we are using to have a shared, quantitative modeling oriented, reference is \textcite{engle-2016}.
Fundamental factor models
Three-factor model
This model was introduced by \textcite{fama-french-1993}, expanding the work begun in \textcite{fama-french-1992}. In their cross-sectional analysis, the authors use the method of \textcite{black-jensen-scholes-1972}. However, the illustration \textcite{engle-2016} do of the model applies the usual \textcite{fama-macbeth-1973} regression analysis. I suspect this difference is due to the fact that, whereas the former work is aimed at inferring and comparing common premia between both stocks and bonds, the latter is focused on providing an analysis on stock returns.
Following \textcite{engle-2016}, we express the so called Fama-French three-factor model as
$$
r_{p,t} = \beta_{0} + \beta_{1, p}MKT_{t} + \beta_{2, p}SMB_{t} + \beta_{3, p}HML_{t} + \epsilon_{t}
$$
where $r_{p,t}$ is the excess return of the portfolio $p$ over the time period $t$ (or, similarly, $r_{i,t}$ for a given security); $MKT_{t}$, $SMB_{t}$, and $HML_{t}$ are the returns of the "market", "small minus big" and "high minus low" factor mimicking portfolios, respectively. The intercept $\beta_{0}$ captures the average excess return that remain unexplained by the factors considered, whereas the remaining betas represent the sensitivities with respect to the each corresponding factor. The variable $\epsilon_{t}$ is the error term.
Estimating factor models
# Following suggestions, I'm working on reproducing models with existing functionality first.
# In particular, the 'FactorAnalytics' package has been suggest as useful. Which likely it is.
# However, I'm encountering some small issues in first attempts.
#
# TODO:
# There are many aspects overlooked here. They must be included in our implementations
# to guarantee analysis accuracy and usage flexibility (e.g. data winsorization).
# The examples below are first attempts to get the mechanics of some 'FactorAnalytics'
# functions and evaluate their feasibility for our purposes.
#
# install_github('FactorAnalytics')
library(ExpectedReturns)
library(FactorAnalytics)
library(xts)
### Experimenting fitFfm() to fit FF three-factor model ###
## Get and set data to pass to fitFfm()
data("factorDataSetDjia5Yrs")
data("FF3.monthly")
# select some variables needed
factorDataSetDjia5Yrs <- factorDataSetDjia5Yrs[, c('DATE', 'TICKER', 'NAME', 'RETURN.OLD', 'RETURN', 'RETURN.DIFF')]
# make 'ff3.data.monthly' and 'factorDataSetDjia5Yrs' compatible for testing
ff3.data.monthly.dates <- index(FF3.monthly)
ff3.data.monthly <- as.data.frame(coredata(FF3.monthly))
ff3.data.monthly <- cbind(ff3.data.monthly.dates, ff3.data.monthly)
colnames(ff3.data.monthly)[1] <- 'DATE'
factorDataSetDjia5Yrs$DATE <- as.Date(factorDataSetDjia5Yrs$DATE)
test.data <- merge(factorDataSetDjia5Yrs, ff3.data.monthly)
# excess returns
test.data$EXC.RETURN <- test.data$RETURN - test.data$RF # FF risk-free rate (one-month Treasury bill rate from Ibbotson Associates)
test.data$RF <- NULL
Fama-French
# factor exposures
exposure.vars <- c('MKT.RF', 'SMB', 'HML')
## Run fitFfm() to fit factor model using cross-sectional regression
fit.test <- fitFfm(data=test.data, asset.var="TICKER", ret.var="EXC.RETURN", date.var="DATE",
exposure.vars=exposure.vars, addIntercept=TRUE)
fit.test
Fama-MacBeth
#
# TODO:
# Again, there are aspects overlooked. Both in the pre-processing (winsorization,
# truncation, etc.) and in the post-processing of the Fama-MacBeth methodology
# (standard error corrections, test statistics adjustements and so on). Then their
# summaries as well.
# Once the core method is in place they are both straightforward.
#
# Also, it's rather naive in this form. However, it can be more or less easily
# improved and extended to other models.
# Merge & match FF3 factors data and stocks data
#test.data <- merge(factorDataSetDjia5Yrs, ff3.data.monthly)
## Fama-MacBeth for FF Three-factor model, looping over lm()
# test.data$EXC.RETURN <- test.data$RETURN - test.data$RF # FF risk-free rate (one-month Treasury bill rate from Ibbotson Associates)
periods <- unique(factorDataSetDjia5Yrs$DATE)
tickers <- unique(factorDataSetDjia5Yrs$TICKER)
factor.vars <- c('MKT.RF', 'SMB', 'HML')
# Run N time-series regressions
beta.coefs <- matrix(NA, length(tickers), length(factor.vars) + 1)
for (i in 1:length(tickers)) {
asset.reg <- lm(EXC.RETURN ~ MKT.RF + SMB + HML, data=test.data[which(test.data[, 'TICKER'] == tickers[i]), ])
beta.coefs[i, ] <- asset.reg$coefficients
}
row.names(beta.coefs) <- tickers
colnames(beta.coefs) <- c('(Intercept)', 'MKT.RF', 'SMB', 'HML')
beta.coefs
# Run T cross-sectional regressions
gamma.coefs <- matrix(NA, length(periods), length(factor.vars) + 1)
gamma.rsq <- matrix(NA, length(periods), 1)
for (t in 1:length(periods)) {
per.period.assets.ret <- test.data[which(test.data[, 'DATE'] == periods[t]), 'EXC.RETURN']
data <- data.frame('EXC.RETURN'=per.period.assets.ret, beta.coefs[, 2:4]) # excluding prev. est. intercepts
time.reg <- lm(EXC.RETURN ~ MKT.RF + SMB + HML, data=data)
gamma.coefs[t, ] <- time.reg$coefficients
}
gamma.coefs <- data.frame(periods, gamma.coefs)
colnames(gamma.coefs) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML')
gamma.coefs
# Average last regression coefficients estimates
avg.coefs <- colMeans(gamma.coefs[, 2:ncol(gamma.coefs)])
avg.coefs
Fama-Macbeth plm
# NOTE:
# same limitations stated in previous TODOs still hold.
# Only concerning with the core mechanics so far.
library(plm) # to estimate model
# match and merge FF3 factors data and stocks data
test.data.plm <- test.data
# Indexes to determine panel structure
# 'PERIOD.ID', each date gets same number id across
periods <- unique(factorDataSetDjia5Yrs$DATE)
periods.id <- 1:length(periods)
periods.id <- data.frame('DATE'=periods, 'PERIOD.ID'=periods.id)
test.data.plm <- merge(test.data.plm, periods.id, by='DATE')
# 'ASSET.ID', each asset gets same number id across
tickers <- unique(factorDataSetDjia5Yrs$TICKER)
assets.id <- 1:length(tickers)
assets.id <- data.frame('TICKER'=tickers, 'ASSET.ID'=assets.id)
test.data.plm <- merge(test.data.plm, assets.id) # this messes up dates order within a ticker
# Sort data set
test.data.plm <- test.data.plm[order(test.data.plm[, 'ASSET.ID'], test.data.plm[, 'PERIOD.ID']), ]
row.names(test.data.plm) <- NULL
#test.data.plm$EXC.RETURN <- test.data.plm$RETURN - test.data.plm$RF # FF risk-free rate (1-month T-bill rate)
test.data.input <- test.data.plm[, c("ASSET.ID", "PERIOD.ID", "DATE", "EXC.RETURN", "MKT.RF", "SMB", "HML")]
# Time-series regressions
fm.ts.reg <- plm::pmg(EXC.RETURN ~ MKT.RF + SMB + HML,
data=test.data.input, index=c('ASSET.ID', 'PERIOD.ID'))
summary(fm.ts.reg)
betas <- t(fm.ts.reg$indcoef) # all coefficients
rownames(betas) <- tickers
colnames(betas) <- paste('BETA', colnames(betas), sep='.')
betas
# (Check my Fama-MacBeth snippet vs. `plm` version)
betas == beta.coefs
# Cross-sectional regressions
test.data.input.second <- test.data.input[order(test.data.input[, 'DATE']), ]
test.data.input.second <- data.frame(
test.data.input.second[, c("PERIOD.ID", "ASSET.ID", "DATE", "EXC.RETURN")],
betas[, 2:4]
)
fm.cs.reg <- plm::pmg(EXC.RETURN ~ BETA.MKT.RF + BETA.SMB + BETA.HML,
data=test.data.input.second, index=c('PERIOD.ID', 'ASSET.ID'))
summary(fm.cs.reg)
gammas <- t(fm.cs.reg$indcoef) # all coefficients
gammas <- data.frame(periods, gammas)
colnames(gammas) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML')
gammas
# (Check my Fama-MacBeth snippet vs. `plm` version)
gammas == gamma.coefs
#refactor by Jiarui
#Use fitTsfm to do time series regression
test.data.third <- data.table::dcast(test.data, DATE ~ TICKER, value.var = "EXC.RETURN")
test.data.third <- merge(test.data.third, ff3.data.monthly[c('DATE', 'MKT.RF','SMB','HML')], by='DATE')
row.names(test.data.third) <- test.data.third$DATE
test.data.third$DATE <- NULL
fit.test <- fitTsfm(asset.names = unique(factorDataSetDjia5Yrs$TICKER), factor.names = c('MKT.RF','SMB','HML'), data = test.data.third)
fit.test.coefs <- cbind(fit.test$alpha, fit.test$beta)
fit.test.coefs == betas
fit.test.coefs == beta.coefs
#Use fitFfm to do cross sectional regression
test.data.fourth <- data.frame(test.data[, c("DATE", "TICKER", "EXC.RETURN")], fit.test.coefs[, 2:4])
exposure.vars <- c('MKT.RF', 'SMB', 'HML')
fit.test.second <- fitFfm(data=test.data.fourth, asset.var="TICKER", ret.var="EXC.RETURN", date.var="DATE", exposure.vars=exposure.vars, addIntercept=TRUE, lagExposures = FALSE)
fit.test.second.coefs <- data.frame(unique(test.data.fourth$DATE), fit.test.second$factor.returns)
colnames(fit.test.second.coefs) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML')
row.names(fit.test.second.coefs) <- NULL
fit.test.second.coefs == gammas
fit.test.second.coefs == gamma.coefs
JustinMShea/ExpectedReturns documentation built on June 28, 2024, 5:37 p.m.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>", echo = TRUE )
# Mentors, please feel free to add yourself to the authors' field if you wish.
Introduction
Our main reference is \textcite{ilmanen-2011} in that we want to reproduce selected key results the author discusses on factor-based investing. However, to achieve those results many more studies from the academic and practitioners' literature need to be implemented first. An excellent resource we are using to have a shared, quantitative modeling oriented, reference is \textcite{engle-2016}.
Fundamental factor models
Three-factor model
This model was introduced by \textcite{fama-french-1993}, expanding the work begun in \textcite{fama-french-1992}. In their cross-sectional analysis, the authors use the method of \textcite{black-jensen-scholes-1972}. However, the illustration \textcite{engle-2016} do of the model applies the usual \textcite{fama-macbeth-1973} regression analysis. I suspect this difference is due to the fact that, whereas the former work is aimed at inferring and comparing common premia between both stocks and bonds, the latter is focused on providing an analysis on stock returns.
Following \textcite{engle-2016}, we express the so called Fama-French three-factor model as $$ r_{p,t} = \beta_{0} + \beta_{1, p}MKT_{t} + \beta_{2, p}SMB_{t} + \beta_{3, p}HML_{t} + \epsilon_{t} $$ where $r_{p,t}$ is the excess return of the portfolio $p$ over the time period $t$ (or, similarly, $r_{i,t}$ for a given security); $MKT_{t}$, $SMB_{t}$, and $HML_{t}$ are the returns of the "market", "small minus big" and "high minus low" factor mimicking portfolios, respectively. The intercept $\beta_{0}$ captures the average excess return that remain unexplained by the factors considered, whereas the remaining betas represent the sensitivities with respect to the each corresponding factor. The variable $\epsilon_{t}$ is the error term.
Estimating factor models
# Following suggestions, I'm working on reproducing models with existing functionality first. # In particular, the 'FactorAnalytics' package has been suggest as useful. Which likely it is. # However, I'm encountering some small issues in first attempts. # # TODO: # There are many aspects overlooked here. They must be included in our implementations # to guarantee analysis accuracy and usage flexibility (e.g. data winsorization). # The examples below are first attempts to get the mechanics of some 'FactorAnalytics' # functions and evaluate their feasibility for our purposes. # # install_github('FactorAnalytics') library(ExpectedReturns) library(FactorAnalytics) library(xts) ### Experimenting fitFfm() to fit FF three-factor model ### ## Get and set data to pass to fitFfm() data("factorDataSetDjia5Yrs") data("FF3.monthly") # select some variables needed factorDataSetDjia5Yrs <- factorDataSetDjia5Yrs[, c('DATE', 'TICKER', 'NAME', 'RETURN.OLD', 'RETURN', 'RETURN.DIFF')] # make 'ff3.data.monthly' and 'factorDataSetDjia5Yrs' compatible for testing ff3.data.monthly.dates <- index(FF3.monthly) ff3.data.monthly <- as.data.frame(coredata(FF3.monthly)) ff3.data.monthly <- cbind(ff3.data.monthly.dates, ff3.data.monthly) colnames(ff3.data.monthly)[1] <- 'DATE' factorDataSetDjia5Yrs$DATE <- as.Date(factorDataSetDjia5Yrs$DATE) test.data <- merge(factorDataSetDjia5Yrs, ff3.data.monthly) # excess returns test.data$EXC.RETURN <- test.data$RETURN - test.data$RF # FF risk-free rate (one-month Treasury bill rate from Ibbotson Associates) test.data$RF <- NULL
Fama-French
# factor exposures exposure.vars <- c('MKT.RF', 'SMB', 'HML') ## Run fitFfm() to fit factor model using cross-sectional regression fit.test <- fitFfm(data=test.data, asset.var="TICKER", ret.var="EXC.RETURN", date.var="DATE", exposure.vars=exposure.vars, addIntercept=TRUE) fit.test
Fama-MacBeth
# # TODO: # Again, there are aspects overlooked. Both in the pre-processing (winsorization, # truncation, etc.) and in the post-processing of the Fama-MacBeth methodology # (standard error corrections, test statistics adjustements and so on). Then their # summaries as well. # Once the core method is in place they are both straightforward. # # Also, it's rather naive in this form. However, it can be more or less easily # improved and extended to other models. # Merge & match FF3 factors data and stocks data #test.data <- merge(factorDataSetDjia5Yrs, ff3.data.monthly) ## Fama-MacBeth for FF Three-factor model, looping over lm() # test.data$EXC.RETURN <- test.data$RETURN - test.data$RF # FF risk-free rate (one-month Treasury bill rate from Ibbotson Associates) periods <- unique(factorDataSetDjia5Yrs$DATE) tickers <- unique(factorDataSetDjia5Yrs$TICKER) factor.vars <- c('MKT.RF', 'SMB', 'HML') # Run N time-series regressions beta.coefs <- matrix(NA, length(tickers), length(factor.vars) + 1) for (i in 1:length(tickers)) { asset.reg <- lm(EXC.RETURN ~ MKT.RF + SMB + HML, data=test.data[which(test.data[, 'TICKER'] == tickers[i]), ]) beta.coefs[i, ] <- asset.reg$coefficients } row.names(beta.coefs) <- tickers colnames(beta.coefs) <- c('(Intercept)', 'MKT.RF', 'SMB', 'HML') beta.coefs # Run T cross-sectional regressions gamma.coefs <- matrix(NA, length(periods), length(factor.vars) + 1) gamma.rsq <- matrix(NA, length(periods), 1) for (t in 1:length(periods)) { per.period.assets.ret <- test.data[which(test.data[, 'DATE'] == periods[t]), 'EXC.RETURN'] data <- data.frame('EXC.RETURN'=per.period.assets.ret, beta.coefs[, 2:4]) # excluding prev. est. intercepts time.reg <- lm(EXC.RETURN ~ MKT.RF + SMB + HML, data=data) gamma.coefs[t, ] <- time.reg$coefficients } gamma.coefs <- data.frame(periods, gamma.coefs) colnames(gamma.coefs) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML') gamma.coefs # Average last regression coefficients estimates avg.coefs <- colMeans(gamma.coefs[, 2:ncol(gamma.coefs)]) avg.coefs
Fama-Macbeth plm
# NOTE: # same limitations stated in previous TODOs still hold. # Only concerning with the core mechanics so far. library(plm) # to estimate model # match and merge FF3 factors data and stocks data test.data.plm <- test.data # Indexes to determine panel structure # 'PERIOD.ID', each date gets same number id across periods <- unique(factorDataSetDjia5Yrs$DATE) periods.id <- 1:length(periods) periods.id <- data.frame('DATE'=periods, 'PERIOD.ID'=periods.id) test.data.plm <- merge(test.data.plm, periods.id, by='DATE') # 'ASSET.ID', each asset gets same number id across tickers <- unique(factorDataSetDjia5Yrs$TICKER) assets.id <- 1:length(tickers) assets.id <- data.frame('TICKER'=tickers, 'ASSET.ID'=assets.id) test.data.plm <- merge(test.data.plm, assets.id) # this messes up dates order within a ticker # Sort data set test.data.plm <- test.data.plm[order(test.data.plm[, 'ASSET.ID'], test.data.plm[, 'PERIOD.ID']), ] row.names(test.data.plm) <- NULL #test.data.plm$EXC.RETURN <- test.data.plm$RETURN - test.data.plm$RF # FF risk-free rate (1-month T-bill rate) test.data.input <- test.data.plm[, c("ASSET.ID", "PERIOD.ID", "DATE", "EXC.RETURN", "MKT.RF", "SMB", "HML")] # Time-series regressions fm.ts.reg <- plm::pmg(EXC.RETURN ~ MKT.RF + SMB + HML, data=test.data.input, index=c('ASSET.ID', 'PERIOD.ID')) summary(fm.ts.reg) betas <- t(fm.ts.reg$indcoef) # all coefficients rownames(betas) <- tickers colnames(betas) <- paste('BETA', colnames(betas), sep='.') betas # (Check my Fama-MacBeth snippet vs. `plm` version) betas == beta.coefs # Cross-sectional regressions test.data.input.second <- test.data.input[order(test.data.input[, 'DATE']), ] test.data.input.second <- data.frame( test.data.input.second[, c("PERIOD.ID", "ASSET.ID", "DATE", "EXC.RETURN")], betas[, 2:4] ) fm.cs.reg <- plm::pmg(EXC.RETURN ~ BETA.MKT.RF + BETA.SMB + BETA.HML, data=test.data.input.second, index=c('PERIOD.ID', 'ASSET.ID')) summary(fm.cs.reg) gammas <- t(fm.cs.reg$indcoef) # all coefficients gammas <- data.frame(periods, gammas) colnames(gammas) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML') gammas # (Check my Fama-MacBeth snippet vs. `plm` version) gammas == gamma.coefs #refactor by Jiarui #Use fitTsfm to do time series regression test.data.third <- data.table::dcast(test.data, DATE ~ TICKER, value.var = "EXC.RETURN") test.data.third <- merge(test.data.third, ff3.data.monthly[c('DATE', 'MKT.RF','SMB','HML')], by='DATE') row.names(test.data.third) <- test.data.third$DATE test.data.third$DATE <- NULL fit.test <- fitTsfm(asset.names = unique(factorDataSetDjia5Yrs$TICKER), factor.names = c('MKT.RF','SMB','HML'), data = test.data.third) fit.test.coefs <- cbind(fit.test$alpha, fit.test$beta) fit.test.coefs == betas fit.test.coefs == beta.coefs #Use fitFfm to do cross sectional regression test.data.fourth <- data.frame(test.data[, c("DATE", "TICKER", "EXC.RETURN")], fit.test.coefs[, 2:4]) exposure.vars <- c('MKT.RF', 'SMB', 'HML') fit.test.second <- fitFfm(data=test.data.fourth, asset.var="TICKER", ret.var="EXC.RETURN", date.var="DATE", exposure.vars=exposure.vars, addIntercept=TRUE, lagExposures = FALSE) fit.test.second.coefs <- data.frame(unique(test.data.fourth$DATE), fit.test.second$factor.returns) colnames(fit.test.second.coefs) <- c('DATE', '(Intercept)', 'MKT.RF', 'SMB', 'HML') row.names(fit.test.second.coefs) <- NULL fit.test.second.coefs == gammas fit.test.second.coefs == gamma.coefs
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