# Load necessary libraries
library(tidyquant)
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## Loading required package: quantmod
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library(lubridate)
library(timetk)
library(purrr)
library(quantmod)
library(dplyr)
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library(tidyr)
# Define the ETF symbols
symbols <- c("SPY", "QQQ", "EEM", "IWM", "EFA", "TLT", "IYR", "GLD")
# Download the data from Yahoo Finance
prices <- getSymbols(symbols, src = 'yahoo', from = "2010-01-01", auto.assign = TRUE, warnings = FALSE) %>%
map(~ Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
# Convert xts to tibble
prices_tibble <- tk_tbl(prices, rename_index = "date")
# Ensure date column is in character format, then convert to Date type
prices_tibble <- prices_tibble %>%
mutate(date = as.character(date)) %>%
mutate(date = as.Date(date))
# Calculate weekly log returns
weekly_prices <- prices_tibble %>%
group_by(week = floor_date(date, "week")) %>%
summarize(across(all_of(symbols), last))
weekly_returns <- weekly_prices %>%
mutate(across(all_of(symbols), ~ log(. / lag(.)))) %>%
drop_na()
# Calculate monthly log returns
monthly_prices <- prices_tibble %>%
group_by(month = floor_date(date, "month")) %>%
summarize(across(all_of(symbols), last))
monthly_returns <- monthly_prices %>%
mutate(across(all_of(symbols), ~ log(. / lag(.)))) %>%
drop_na()
# Convert to tibble format
monthly_returns_tibble <- as_tibble(monthly_returns)
# Rename 'month' column to 'date'
monthly_returns_tibble <- monthly_returns_tibble %>%
rename(date = month)
# Load the Fama/French data from URL
FFdata <- read.csv("https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Research_Data_Factors.CSV", skip = 3)
# Clean and format the data, handling parsing errors
FFdata <- FFdata %>%
mutate(date = parse_date_time(X, orders = "Y%m", quiet = TRUE)) %>%
filter(!is.na(date)) %>%
select(date, Mkt.RF, SMB, HML, RF)
# Convert character columns to numeric
FFdata <- FFdata %>%
mutate(across(-date, ~ as.numeric(gsub(" ", "", .))))
# Ensure date columns are present and in Date format
FFdata <- FFdata %>%
mutate(date = as.Date(date))
# Merge the ETF returns with the FF factors data
merged_data <- left_join(monthly_returns_tibble, FFdata, by = 'date')
# Extract the relevant returns for the CAPM model
capm_returns <- merged_data %>%
filter(date >= "2010-02-01" & date <= "2015-01-31") %>%
select(-date, -Mkt.RF, -SMB, -HML, -RF)
# Compute the covariance matrix
capm_cov_matrix <- cov(capm_returns)
# Extract the relevant returns and FF factors for the FF model
ff_returns <- merged_data %>%
filter(date >= "2010-02-01" & date <= "2015-01-31")
# Compute the residuals for each ETF by regressing on the FF factors
residuals <- ff_returns %>%
select(all_of(symbols)) %>%
map_dfc(~ residuals(lm(. ~ Mkt.RF + SMB + HML, data = ff_returns)))
# Compute the covariance matrix of the residuals
ff_cov_matrix <- cov(residuals)
# Function to compute GMV portfolio weights
compute_gmv_weights <- function(cov_matrix) {
inv_cov <- solve(cov_matrix)
ones <- rep(1, nrow(inv_cov))
weights <- inv_cov %*% ones / sum(inv_cov %*% ones)
return(weights)
}
# Compute the weights
gmv_weights_capm <- compute_gmv_weights(capm_cov_matrix)
gmv_weights_ff <- compute_gmv_weights(ff_cov_matrix)
# Print weights for verification
print(gmv_weights_capm)
## [,1]
## SPY 0.94116583
## QQQ -0.18088331
## EEM -0.01437082
## IWM -0.05252841
## EFA -0.01342445
## TLT 0.48106300
## IYR -0.21173959
## GLD 0.05071774
print(gmv_weights_ff)
## [,1]
## SPY 0.8299846916
## QQQ 0.0052295955
## EEM 0.0001153044
## IWM 0.1868628046
## EFA 0.0009328018
## TLT 0.0032729219
## IYR -0.0286049238
## GLD 0.0022068040
# Function to compute cumulative returns from weights and returns
compute_cumulative_returns <- function(returns, weights) {
portfolio_returns <- rowSums(returns * weights)
cumulative_returns <- cumprod(1 + portfolio_returns) - 1
return(cumulative_returns)
}
# Prepare returns data for backtesting
returns_data <- merged_data %>%
filter(date > as.Date("2015-01-31")) %>%
select(date, all_of(symbols)) %>%
mutate(across(where(is.numeric), ~exp(.) - 1)) %>% # Convert log returns back to normal returns
tk_tbl(rename_index = "date")
## Warning in tk_tbl.data.frame(., rename_index = "date"): Warning: No index to
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# Backtest using CAPM model
cum_returns_capm <- compute_cumulative_returns(returns_data %>% select(-date), gmv_weights_capm)
# Backtest using Fama-French model
cum_returns_ff <- compute_cumulative_returns(returns_data %>% select(-date), gmv_weights_ff)
# Plot cumulative returns
plot(as.Date(returns_data$date), cum_returns_capm, type = "l", col = "blue", xlab = "Time", ylab = "Cumulative Returns", main = "GMV Portfolio Performance")
lines(as.Date(returns_data$date), cum_returns_ff, col = "red")
legend("topright", legend = c("CAPM", "Fama-French"), col = c("blue", "red"), lty = 1)
