Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Seminar 6
Content:
• CAPM and FF3 estimates comparison
• Long-short portfolios construction
CAPM and FF3 estimates comparison
We switch to “dataset_2.Rdata”, which contains 62 stocks
from the NASDAQ 100 index, as we need more stocks for a
meaningful long-short portfolio.
First we use the entire dataset to compare the expected
returns estimated by CAPM and by FF3.
We start with the CAPM.
With a loop we estimate all the betas using the regression
𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 𝑅 𝑀𝑘𝑡 − 𝑅𝑓
Then we compute the expected returns as
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖(𝐸 𝑅 𝑚 − 𝑅𝑓)
CAPM and FF3 estimates comparison
Now we use the Fama-French three-factor model.
With a first loop we estimate all the loadings using this
regression, which is fitted separately for all the assets
𝑅𝑖𝑡 − 𝑅𝑓𝑡 = 𝛼𝑖 + 𝑏𝑖1 𝑀𝐾𝑇𝑡 + 𝑏𝑖2 𝑆𝑀𝐵𝑡 + 𝑏𝑖3 𝐻𝑀𝐿 𝑡
Then we estimate the premia with a second loop that for
each period fits this regression:
𝑅𝑖 − 𝑅𝑓 = 𝛾𝑡0 + 𝛾𝑡1
෢𝑏𝑖1 + 𝛾𝑡2
෢𝑏𝑖2 + 𝛾𝑡3
෢𝑏𝑖3
We now compute the expected return of each asset 𝑖 as:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝛾 𝑀𝐾𝑇 + 𝑏𝑖2 𝛾𝑆𝑀𝐵 + 𝑏𝑖3 𝛾 𝐻𝑀𝐿
Long-short portfolios construction
Now we use the Fama-French three-factor model.
We now use the same estimation techniques, but we do it
on a 120-month rolling window.
At each period we get an estimate of the expected returns.
We want to go long on the 20 stocks with the highest
expected return, and short on the 20 with the lowest.
The function “ls_weights” assigns weight 1/20 to the top 20
stocks, -1/20 to the bottom 20 stocks, and 0 otherwise.
We then compute the portfolio out-of-sample returns.