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library(quantmod) library(PerformanceAnalytics) library(tseries) library(ggplot2) library(xts) library(dplyr)

Load data

data <- read.csv(“myetf4.csv”) data\(Date <- as.Date(data\)Date, format=“%Y-%m-%d”)

Convert to xts format

prices <- xts(data[, -1], order.by = data$Date)

Compute daily returns

daily_returns <- ROC(prices, type=“discrete”)[-1, ]

Compute covariance matrix

cov_matrix <- cov(daily_returns, use = “complete.obs”)

GMVP Calculation

inv_cov <- solve(cov_matrix) one_vec <- rep(1, ncol(inv_cov)) w_gmvp <- inv_cov %% one_vec / sum(inv_cov %% one_vec)

GMVP Return and Standard Deviation

gmvp_return <- colMeans(daily_returns, na.rm = TRUE) %% w_gmvp gmvp_sd <- sqrt(t(w_gmvp) %% cov_matrix %*% w_gmvp)

Convert to monthly returns

monthly_prices <- to.monthly(prices, indexAt=‘lastof’, OHLC=FALSE) monthly_returns <- ROC(monthly_prices, type=“discrete”)[-1, ]

Compute covariance matrix for monthly returns

cov_matrix_monthly <- cov(monthly_returns, use = “complete.obs”)

GMVP Calculation for monthly returns

inv_cov_monthly <- solve(cov_matrix_monthly) w_gmvp_monthly <- inv_cov_monthly %% one_vec / sum(inv_cov_monthly %% one_vec)

gmvp_return_monthly <- colMeans(monthly_returns, na.rm = TRUE) %% w_gmvp_monthly gmvp_sd_monthly <- sqrt(t(w_gmvp_monthly) %% cov_matrix_monthly %*% w_gmvp_monthly)

Tangency Portfolio Calculation

mu <- colMeans(monthly_returns, na.rm = TRUE) w_tangency <- inv_cov_monthly %% mu / sum(inv_cov_monthly %% mu)

tangency_return <- sum(mu * w_tangency) tangency_sd <- sqrt(t(w_tangency) %% cov_matrix_monthly %% w_tangency)

Results

list( GMVP_Daily = list(weights = w_gmvp, return = gmvp_return, sd = gmvp_sd), GMVP_Monthly = list(weights = w_gmvp_monthly, return = gmvp_return_monthly, sd = gmvp_sd_monthly), Tangency = list(weights = w_tangency, return = tangency_return, sd = tangency_sd) )