Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Seminar 3
Content:
• Compute mean and SD and plot them
• Compute and plot wealth dynamics
• Mean and variance of a portfolio
• Rolling and expanding window estimation
Compute mean and SD and plot them
• We can load the data using “load("dataset.RData")” if it is
in the same working directory. Or we can use the “load
workspace” button in the environment panel on the right.
• First we select only the columns with returns.
• To compute the mean return of each stock, we compute
the mean of each column with the “colMeans” function.
• To compute the standard deviation of each stock we use
“apply”. The “2” indicates we want to apply the function
“sd” by column. If we wrote “1” we would apply it by row.
• We can now draw the scatterplot.
Compute and plot wealth dynamics
• We compute the returns of an equally weighted portfolio
simply by computing the mean return of each row using
“rowMeans”, and we compute and plot its mean and sd.
• We use “points” to plot it on the same graph we created
before. We specify the color red with the “col” option.
• We then create a custom function to compute the
evolution of wealth of this portfolio. It uses the formula:
𝑉𝑇 = 𝑉0 + ෍
𝑡=1
𝑇
( 𝑉𝑡−1 𝑅𝑡)
Compute and plot wealth dynamics
• While the dataset already has a date column, the vector
of wealth has one additional observation, because we
added one with value 1 at the beginning (we invest 1 unit
of wealth one month before the date of the first return).
• So we create a new date vector by creating a series of first
day of month dates, and then subtracting one day. This
gives us a vector of end of month dates.
• Then we plot. The date is used on the horizontal axis and
the wealth on the vertical axis. “type= " l" ” is used to
draw a line; otherwise we would get points.
Mean and variance of a portfolio
• We consider five of the stocks and use them to assemble
a portfolio with given weights.
• We can easily compute the mean of this portfolio using
the formula in matrix notation µ 𝑝 = 𝒘′µ
• Likewise, we compute the variance of the portfolio using
𝜎 𝑝
2
= 𝒘′∑𝒘
• If we want the standard deviation of the portfolio, we can
simply compute the square root of the variance.
Rolling and expanding window estimation
We now compute rolling and expanding window estimates
of the mean and covariance matrix
• We do this using a “for” cycle. To get a rolling window we
index the starting of the subset with “i”, while to get an
expanding window we always start at “1”.
• We store the vectors of means and the covariance
matrices into list objects.