Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Seminar 4
Content:
• Estimation of betas
• Plotting the CML and the SML
Estimation of betas
• 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.
• In the same way we load “SP500.RData”
• To estimate the beta we fit the regression:
𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 𝑅 𝑀𝑘𝑡 − 𝑅𝑓
• We start by considering only the AXP stock
• We fit the regression using the “lm” function
• The alpha and beta are stored in the “coefficients” part of
the object that we used to store the regression model
Estimation of betas
• We plot the regression line using the “abline” function
• Then we do the same with the BA stock
• BA has a higher beta then AXP and, as the theory
predicts, its historical mean return is also higher
• We finally use a “for” loop to compute all the betas
• The beta of the risk-free asset is 0, and the beta of the
market is 1, so there is no need to compute them.
Plotting the CML and the SML
• We compute the expected return of the stocks using the
CAPM formula:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖(𝐸 𝑅 𝑚 − 𝑅𝑓)
• The expected return of the T-bill (“risk-free asset”) and
of the S&P 500 (“market portfolio”) are given by their
respective mean return
• We compute the standard deviation of all the stocks, the
market and the risk-free asset (the latter should ideally
have zero risk, but the T-bill rate, which we use as proxy,
has some variability)
Plotting the CML and the SML
• We plot the expected returns against the standard
deviations
• We draw a line that connects the T-bill with the S&P
500 portfolio. This is the Capital Market Line (CML)
• The we plot another graph with the betas instead of
the standard deviation.
• We draw again a line that connects the T-bill with the
S&P 500 portfolio. This is now the Security Market Line
(SML)