library(tidyquant)
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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')
# Check the first few rows of the merged data
head(merged_data)
## # A tibble: 6 × 13
## date SPY QQQ EEM IWM EFA TLT IYR GLD
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2010-02-01 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 2 2010-03-01 0.0591 0.0743 0.0780 0.0791 0.0619 -0.0208 0.0930 -0.00440
## 3 2010-04-01 0.0154 0.0222 -0.00166 0.0552 -0.0284 0.0327 0.0619 0.0572
## 4 2010-05-01 -0.0828 -0.0768 -0.0986 -0.0784 -0.119 0.0498 -0.0585 0.0301
## 5 2010-06-01 -0.0531 -0.0616 -0.0141 -0.0806 -0.0208 0.0564 -0.0478 0.0233
## 6 2010-07-01 0.0661 0.0701 0.104 0.0651 0.110 -0.00951 0.0899 -0.0522
## # ℹ 4 more variables: Mkt.RF <dbl>, SMB <dbl>, HML <dbl>, RF <dbl>
# 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)
print(capm_cov_matrix)
## SPY QQQ EEM IWM EFA
## SPY 0.0013789602 0.0014370239 0.0017213312 0.0017634400 0.0015961621
## QQQ 0.0014370239 0.0017632586 0.0017694533 0.0018116853 0.0016582372
## EEM 0.0017213312 0.0017694533 0.0033901692 0.0023127599 0.0024955552
## IWM 0.0017634400 0.0018116853 0.0023127599 0.0026629307 0.0019648618
## EFA 0.0015961621 0.0016582372 0.0024955552 0.0019648618 0.0024253981
## TLT -0.0009663047 -0.0009584933 -0.0011821330 -0.0013080775 -0.0011039450
## IYR 0.0011816490 0.0012041824 0.0017966332 0.0015876825 0.0015471681
## GLD 0.0002043559 0.0003904136 0.0009310439 0.0005394847 0.0004417093
## TLT IYR GLD
## SPY -0.0009663047 0.0011816490 0.0002043559
## QQQ -0.0009584933 0.0012041824 0.0003904136
## EEM -0.0011821330 0.0017966332 0.0009310439
## IWM -0.0013080775 0.0015876825 0.0005394847
## EFA -0.0011039450 0.0015471681 0.0004417093
## TLT 0.0015464531 -0.0003632219 0.0001797642
## IYR -0.0003632219 0.0019945702 0.0005353308
## GLD 0.0001797642 0.0005353308 0.0029025882
# 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)
print(ff_cov_matrix)
## SPY QQQ EEM IWM EFA
## SPY 3.259887e-06 3.623086e-06 1.782721e-06 8.351961e-07 1.903076e-06
## QQQ 3.623086e-06 2.067122e-04 -5.274492e-05 -1.248084e-05 -3.610020e-05
## EEM 1.782721e-06 -5.274492e-05 1.213625e-03 4.639351e-05 5.052328e-04
## IWM 8.351961e-07 -1.248084e-05 4.639351e-05 1.656390e-05 3.183174e-05
## EFA 1.903076e-06 -3.610020e-05 5.052328e-04 3.183174e-05 5.328003e-04
## TLT 8.488490e-06 -4.926446e-06 1.964150e-05 2.785429e-05 -2.701055e-05
## IYR 2.136832e-05 -1.638254e-05 3.328167e-04 5.932411e-05 2.082024e-04
## GLD -8.118608e-06 5.140452e-05 5.668764e-04 5.843993e-05 1.954885e-04
## TLT IYR GLD
## SPY 8.488490e-06 2.136832e-05 -8.118608e-06
## QQQ -4.926446e-06 -1.638254e-05 5.140452e-05
## EEM 1.964150e-05 3.328167e-04 5.668764e-04
## IWM 2.785429e-05 5.932411e-05 5.843993e-05
## EFA -2.701055e-05 2.082024e-04 1.954885e-04
## TLT 7.800141e-04 4.560207e-04 2.593810e-04
## IYR 4.560207e-04 1.008806e-03 3.079603e-04
## GLD 2.593810e-04 3.079603e-04 2.510801e-03
# 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(gmv_weights_capm)
## [,1]
## SPY 0.94117621
## QQQ -0.18088876
## EEM -0.01436796
## IWM -0.05253206
## EFA -0.01342691
## TLT 0.48106249
## IYR -0.21174261
## GLD 0.05071960
print(gmv_weights_ff)
## [,1]
## SPY 0.8299984034
## QQQ 0.0052245323
## EEM 0.0001182274
## IWM 0.1868559875
## EFA 0.0009298815
## TLT 0.0032721623
## IYR -0.0286070783
## GLD 0.0022078840