Importing packages required
We will use several packages that will help us pull stocks data and to visualize simulated risks and returns. The following code installs the packages required if they’re not yet available. Afterwards, they will be loaded in our current R session.
## (1) Define the packages that will be needed
packages <- c('quantmod', 'ggplot2', 'dplyr')
## (2) Install them if not yet installed
installed_packages <- packages %in% rownames(installed.packages())
if (any(installed_packages == FALSE)) {
install.packages(packages[!installed_packages])
}
## (3) Load the packages into R session
invisible(lapply(packages, library, character.only = TRUE))Getting adjusted stock prices data
Let’s now load the stocks data that we will need. First, we will define a variable called portfolio that has the stock symbols that we want. Then use lapply to load the weekly stock data of each symbol into R and store them as a list. Afterwards, we’ll only take the adjusted prices, then merge all of them into an xts object.
## Create a character vector that has the stock codes we need
portfolio <- c('AAPL', 'MSFT', 'GOOG', 'AMZN', 'JNJ')
## Load the stocks needed into R
portfolio <- lapply(portfolio, function(x) {getSymbols(
x, periodicity='weekly', auto.assign=FALSE)})
## Get adjusted prices of all stocks
portfolio_adjusted <- lapply(portfolio, Ad)
## Transform into xts
portfolio_adjusted <- do.call(merge, portfolio_adjusted)
## View the first few rows
head(portfolio_adjusted)## AAPL.Adjusted MSFT.Adjusted GOOG.Adjusted AMZN.Adjusted JNJ.Adjusted
## 2007-01-01 2.600933 21.62418 242.6853 38.37 42.64668
## 2007-01-08 2.893595 22.76959 251.5571 38.20 42.65948
## 2007-01-15 2.706438 22.69663 243.9606 37.02 43.37643
## 2007-01-22 2.611025 22.32455 246.9942 36.85 42.29460
## 2007-01-29 2.591759 22.02544 239.8510 37.39 42.62106
## 2007-02-05 2.546499 21.14266 230.0826 38.72 41.99372Getting weekly log returns
We’ll now take the weekly log returns of each symbol.
## Make a list that contains log weekly returns of each stock
portfolio_adjusted <- lapply(portfolio_adjusted, weeklyReturn, type='log')
## Transform into an xts object
portfolio_adjusted <- do.call(merge, portfolio_adjusted)
## Adjust the column names
colnames(portfolio_adjusted) <- c('AAPL', 'MSFT', 'GOOG', 'AMZN', 'JNJ')
## Remove first row since these do not have returns
portfolio_adjusted <- portfolio_adjusted[-1]
## View the first few rows of the log returns
head(portfolio_adjusted)## AAPL MSFT GOOG AMZN JNJ
## 2007-01-08 0.106629447 0.051613838 0.03590425 -0.004440336 0.0003002362
## 2007-01-15 -0.066866295 -0.003209463 -0.03066338 -0.031377235 0.0166667512
## 2007-01-22 -0.035890515 -0.016529488 0.01235824 -0.004602743 -0.0252569272
## 2007-01-29 -0.007406069 -0.013488686 -0.02934704 0.014547698 0.0076891967
## 2007-02-05 -0.017617321 -0.040905349 -0.04157947 0.034953026 -0.0148285558
## 2007-02-12 0.018560637 -0.008315604 0.01727824 0.040739370 -0.0013727184Creating variance-covariance matrix
Our goal is to plot efficient frontier of the symbols we’ve defined above. We will simulate 10,000 combinations of random weights to be assigned to each of the symbols. Then, we’ll calculate the risk and returns that correspond to each combination of weights.
To do that, we need to make a variance-covariance matrix since it’ll be part of the calculations. We can get that using the var function in base R.
## Get variance-covariance matrix
var_covar <- var(portfolio_adjusted)
## Print results
var_covar## AAPL MSFT GOOG AMZN JNJ
## AAPL 0.0018814637 0.0006536266 0.0009219622 0.0009696068 0.0003096191
## MSFT 0.0006536266 0.0012047967 0.0007166929 0.0007124102 0.0003549947
## GOOG 0.0009219622 0.0007166929 0.0014842451 0.0009823727 0.0003496576
## AMZN 0.0009696068 0.0007124102 0.0009823727 0.0022446234 0.0002901278
## JNJ 0.0003096191 0.0003549947 0.0003496576 0.0002901278 0.0005356508Creating random weights
We can now make our random weights.
## Set seed for reproducibility
set.seed(123)
## Get 50,000 random uniform numbers
random_numbers <- runif(50000)
## Transform random numbers into matrix to distribute across all symbols
all_weights <- matrix(random_numbers, nrow=10000, ncol=5)
## Add sixth column with just NAs
all_weights <- cbind(all_weights, rep(NA, 10000))
## Add names
colnames(all_weights) <- c('AAPL', 'MSFT', 'GOOG', 'AMZN', 'JNJ', 'total')
## Loop to convert into actual weights
for (i in 1:10000) {
## Get sum of random numbers in each row
all_weights[i, 6] <- sum(all_weights[i, 1:5])
## Get the actual weights of the random numbers
all_weights[i, 1:5] <-
all_weights[i, 1:5] / all_weights[i, 6]
}
## Delete total column
all_weights <- all_weights[, -6]Calculating risks and returns
We can now compute the risks and returns of the portfolio for each 10,000 random weights. The following are the corresponding formula for each metric.
\[ P_{Rt} = \sum P_w * P_{Cov} \]
Where, \(P_Rt\) is our portfolio return, \(P_w\) are the asset weights, then \(P_{Cov}\) is the individual covariance per variable. We get \(P_Rt\) by getting the sum of the product of weights and covariance.
\[ P_{Rs} = \sqrt \sum (X*P_w) * X \]
Where, \(P_Rs\) is our portfolio risk, \(P_w\) are the asset weights, then \(X\) is the variance-covariance matrix. We get \(P_Rs\) by getting the matrix product of \(X\) and \(P_w\) first, then multiply by \(X\), and get the sum. Finally, take the sum’s square root.
## Create column placeholders
portfolio_risk <- rep(NA, 10000)
portfolio_returns <- rep(NA, 10000)
## loop to calculate risk and return per weights
for (i in 1:10000) {
weights <- all_weights[i, ]
portfolio_risk[i] <- sqrt(sum((weights %*% var_covar) * weights))
portfolio_returns[i] <- sum(weights * var_covar[1, ])
}
## Make a data frame to be used for ggplot2
portfolio_df <- data.frame(portfolio_risk, portfolio_returns)Make the efficient frontier plot
For the final step of this demo, let’s plot the risks and returns.
portfolio_df %>%
ggplot(aes(x=portfolio_risk, y=portfolio_returns)) +
geom_point(alpha=0.2) +
theme_minimal() +
labs(
title='Efficient Frontier graph of 5 assets',
subtitle='AAPL, MSFT, GOOG, AMZN, JNJ')