Q1
library(lubridate)
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library(timetk)
library(purrr)
library(quantmod)
## Loading required package: xts
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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))
Q2
# 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()
Q3
# 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)
Q4
# 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))
Q5-9
# 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 0.0013789598 0.0014370249 0.0017213343 0.0017634418 0.0015961650
## QQQ 0.0014370249 0.0017632673 0.0017694591 0.0018116924 0.0016582428
## EEM 0.0017213343 0.0017694591 0.0033901750 0.0023127678 0.0024955584
## IWM 0.0017634418 0.0018116924 0.0023127678 0.0026629407 0.0019648700
## EFA 0.0015961650 0.0016582428 0.0024955584 0.0019648700 0.0024254008
## TLT -0.0009663043 -0.0009584927 -0.0011821267 -0.0013080755 -0.0011039449
## IYR 0.0011816465 0.0012041825 0.0017966298 0.0015876818 0.0015471656
## GLD 0.0002043556 0.0003904112 0.0009310453 0.0005394841 0.0004417083
## TLT IYR GLD
## SPY -0.0009663043 0.0011816465 0.0002043556
## QQQ -0.0009584927 0.0012041825 0.0003904112
## EEM -0.0011821267 0.0017966298 0.0009310453
## IWM -0.0013080755 0.0015876818 0.0005394841
## EFA -0.0011039449 0.0015471656 0.0004417083
## TLT 0.0015464533 -0.0003632137 0.0001797669
## IYR -0.0003632137 0.0019945663 0.0005353306
## GLD 0.0001797669 0.0005353306 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 3.259852e-06 3.622527e-06 1.785119e-06 8.351710e-07 1.903994e-06
## QQQ 3.622527e-06 2.067153e-04 -5.274417e-05 -1.248076e-05 -3.609938e-05
## EEM 1.785119e-06 -5.274417e-05 1.213627e-03 4.639485e-05 5.052315e-04
## IWM 8.351710e-07 -1.248076e-05 4.639485e-05 1.656409e-05 3.183227e-05
## EFA 1.903994e-06 -3.609938e-05 5.052315e-04 3.183227e-05 5.327990e-04
## TLT 8.486802e-06 -4.926609e-06 1.964589e-05 2.785515e-05 -2.701100e-05
## IYR 2.136890e-05 -1.638159e-05 3.328155e-04 5.932453e-05 2.082011e-04
## GLD -8.116272e-06 5.140087e-05 5.668773e-04 5.843916e-05 1.954869e-04
## TLT IYR GLD
## SPY 8.486802e-06 2.136890e-05 -8.116272e-06
## QQQ -4.926609e-06 -1.638159e-05 5.140087e-05
## EEM 1.964589e-05 3.328155e-04 5.668773e-04
## IWM 2.785515e-05 5.932453e-05 5.843916e-05
## EFA -2.701100e-05 2.082011e-04 1.954869e-04
## TLT 7.800183e-04 4.560251e-04 2.593830e-04
## IYR 4.560251e-04 1.008806e-03 3.079616e-04
## GLD 2.593830e-04 3.079616e-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 0.94117435
## QQQ -0.18088470
## EEM -0.01437210
## IWM -0.05253054
## EFA -0.01342503
## TLT 0.48106317
## IYR -0.21174448
## GLD 0.05071933
print(gmv_weights_ff)
## [,1]
## SPY 0.8299928534
## QQQ 0.0052266489
## EEM 0.0001166393
## IWM 0.1868598232
## EFA 0.0009303720
## TLT 0.0032747418
## IYR -0.0286082982
## GLD 0.0022072195