Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 8
Content:
• Long-short portfolios
• Fama-MacBeth regression
• Example with the Fama-French three-factor model
Long-short portfolios
Suppose we have N assets. Investing in a long-short
portfolio involves:
1. Ranking the assets according to their expected return
2. Divide them in two legs:
- a long leg with a given number of assets with the
highest predicted return
- a short leg with a given number of assets with the
lowest predicted return
Typically, equal weights are used within each leg, but
alternatives are of course possible
Long-short portfolios
Advantages of long-short portfolios:
• No optimization procedures required
• Completely or partially self-financing: money obtained
from shorting can be used for the long positions
Disadvantages of long-short portfolios:
• N has to be relatively small, as it is difficult to short large
number of stocks in practice less diversification
• Placing a substantial amount of wealth in short
positions can be risky
Mean-variance utility maximization
How to rank the assets based on their expected return?
Using the sample mean is NOT appropriate.
A factor-based approach is often used.
This is one typical way factor investing is performed.
It generally involves computing expected returns using a
multifactor model.
Mean-variance utility maximization
Remember that the formula of a multifactor model with
𝑘 factors is:
𝑅𝑖 = 𝛼𝑖 + 𝑏𝑖1 𝑓1 + 𝑏𝑖2 𝑓2 + ⋯ + 𝑏𝑖𝑘 𝑓𝑘 + 𝜀𝑖
In practice, the expected return of a stock given a certain
multifactor model is computed as:
𝐸[𝑅𝑖] = 𝛼𝑖 + 𝑏𝑖1 𝛾1 + 𝑏𝑖2 𝛾2 + ⋯ + 𝑏𝑖𝑘 𝑓𝛾 𝑘
where 𝛾 is the factor risk premium.
Fama-MacBeth regression
The loadings and the risk premia are typically estimated
using the Fama-MacBeth regression, which is a two-stage
multiple linear regression.
Consider 𝑁 assets over 𝑇 periods.
In the first stage, the loadings 𝑏𝑖𝑘 are estimated by
regressing the returns of each asset 𝑖 on the 𝑘 factors,
using the entire set of 𝑇 periods:
𝑅1𝑡 = 𝛼1 + 𝑏11 𝑓1𝑡 + 𝑏12 𝑓2𝑡 + ⋯ + 𝑏1𝑘 𝑓𝑘
𝑅2𝑡 = 𝛼2 + 𝑏21 𝑓1𝑡 + 𝑏22 𝑓2𝑡 + ⋯ + 𝑏2𝑘 𝑓𝑘
⋮
𝑅𝑖𝑡 = 𝛼𝑖 + 𝑏𝑖1 𝑓1𝑡 + 𝑏𝑖2 𝑓2𝑡 + ⋯ + 𝑏𝑖𝑘 𝑓𝑘
⋮
𝑅 𝑁𝑡 = 𝛼 𝑁 + 𝑏 𝑁1 𝑓1𝑡 + 𝑏 𝑁2 𝑓2𝑡 + ⋯ + 𝑏 𝑁𝑘 𝑓𝑘𝑡
Fama-MacBeth regression
In the second stage, the estimated loadings are then
used as explanatory variables in a second regression that,
for each period 𝑡, regresses the asset returns of the
entire set of 𝑁 assets:
𝑅𝑖1 = 𝛾10 + 𝛾11
෢𝑏𝑖1 + 𝛾12
෢𝑏𝑖2 + ⋯ + 𝛾1𝑘
෢𝑏𝑖𝑘
𝑅𝑖2 = 𝛾20 + 𝛾21
෢𝑏𝑖1 + 𝛾22
෢𝑏𝑖2 + ⋯ + 𝛾2𝑘
෢𝑏𝑖𝑘
⋮
𝑅𝑖𝑡 = 𝛾𝑡0 + 𝛾𝑡1
෢𝑏𝑖1 + 𝛾𝑡2
෢𝑏𝑖2 + ⋯ + 𝛾𝑡𝑘
෢𝑏𝑖𝑘
⋮
𝑅𝑖𝑇 = 𝛾 𝑇0 + 𝛾 𝑇1
෢𝑏𝑖1 + 𝛾 𝑇2
෢𝑏𝑖2 + ⋯ + 𝛾 𝑇𝑘
෢𝑏𝑖𝑘
Fama-MacBeth regression
The risk premia are time-varying. A common approach is
to compute their average value over the 𝑇 periods.
The expected return of each asset 𝑖 according to the
chosen multifactor model is given by:
𝐸 𝑅𝑖 = 𝑏𝑖1 𝛾1 + 𝑏𝑖2 𝛾2 + ⋯ + 𝑏𝑖𝑘 𝛾 𝑘
Example with the Fama-French three-factor model
For greater clarity, let us consider how this works with the
Fama-French three-factor model:
𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝑅 𝑚 − 𝑅𝑓 + 𝑏𝑖2 𝑆𝑀𝐵 + 𝑏𝑖3 𝐻𝑀𝐿
In practice the expected return of asset 𝑖 is computed as:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝛾 𝑅 𝑚−𝑅 𝑓
+ 𝑏𝑖2 𝛾𝑆𝑀𝐵 + 𝑏𝑖3 𝛾 𝐻𝑀𝐿
We use the Fama-MacBeth regression to estimate the
loadings and the risk premia. Usually, the excess return is
used as dependent variable, to focus on the component of
the return that is dependent on factor exposure.
Example with the Fama-French three-factor model
Therefore, the first stage regression for each asset 𝑖 is:
𝑅𝑖𝑡 − 𝑅𝑓𝑡 = 𝛼𝑖 + 𝑏𝑖1 𝑅 𝑚𝑡 − 𝑅𝑓𝑡 + 𝑏𝑖2 𝑆𝑀𝐵𝑡 + 𝑏𝑖3 𝐻𝑀𝐿 𝑡
To simplify the notation, we indicate 𝑅 𝑚𝑡 − 𝑅𝑓𝑡 as 𝑀𝐾𝑇:
𝑅𝑖𝑡 − 𝑅𝑓𝑡 = 𝛼𝑖 + 𝑏𝑖1 𝑀𝐾𝑇𝑡 + 𝑏𝑖2 𝑆𝑀𝐵𝑡 + 𝑏𝑖3 𝐻𝑀𝐿 𝑡
This regression needs to be carried out separately for each
of the 𝑁 assets.
We can now set up the second stage regression:
𝑅𝑖 − 𝑅𝑓 = 𝛾𝑡0 + 𝛾𝑡1
෢𝑏𝑖1 + 𝛾𝑡2
෢𝑏𝑖2 + 𝛾𝑡3
෢𝑏𝑖3
This regression needs to be carried out separately for each
of the 𝑇 periods in the estimation window.
Example with the Fama-French three-factor model
The second regression gave us 𝑇 values for 𝛾𝑡1, 𝛾𝑡2 and 𝛾𝑡3.
We compute their average in order to have a single value.
We rename the average of 𝛾𝑡1, 𝛾𝑡2 and 𝛾𝑡3 as 𝛾 𝑀𝐾𝑇, 𝛾𝑆𝑀𝐵
and 𝛾 𝐻𝑀𝐿 respectively, for better clarity.
We also compute the average risk-free rate in order to
have a single value for 𝑅𝑓.
We can now compute the expected return of each asset 𝑖:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝛾 𝑀𝐾𝑇 + 𝑏𝑖2 𝛾𝑆𝑀𝐵 + 𝑏𝑖3 𝛾 𝐻𝑀𝐿
Example with the Fama-French three-factor model
It is now straightforward to create the long-short portfolio:
1. Rank the assets according to their expected return, and
take a long position on those positioned in the upper
part of the ranking, and a short position on those in
lower part of the ranking.
2. Each time the portfolio has to be updated, compute
new estimates of the expected returns. For example, if
we want to update the portfolio monthly, we ne need
to repeat the procedure every month, using the up-todate
data.