FIN 5190
Professor Boldin
Zhengwei Liang
December 6, 2012
Portfolio Establishment
Project Overview and Objectives:
The final project is about portfolio establishment. In the real financial world, investors need to
make decisions on what kind of stock they want to buy and how large portion of each individual
stock to form their investment portfolios. The objective of this project is to use investment
knowledge and financial modeling technique to give investors some clues on portfolio forming.
The investment concepts used in the project include firm-specific risk, envelope portfolio,
efficient frontier, global minimum variance portfolio, and Black-Litterman approach. The
techniques used are excel and gretl. The project contained 10 stocks selection, historical data
collection, finding two envelope portfolios, calculation of efficient frontier, check excel results
with gretl programming, computing global minimum variance portfolio, and optimize portfolio
with Black-Litterman approach.
Financial Modeling Technique Applied:
Computing Efficient Frontier of 10 stocks
Efficient frontier is a group of combined risky assets, which is portfolios, that has the best
possible expected return given their risk level. Investors should only choose portfolios on the
efficient frontier to maximize their returns based on certain risk or minimize their risks based on
predetermined returns. The whole idea of computing efficient frontier is to find two envelope
portfolios x and y, which are tangency portfolios on the efficient frontier given a constant c.
These two portfolios would led us to the whole efficient frontier, since all envelope portfolios are
convex combinations of x and y.
In order to compute the efficient frontier, the first step is stock selection. The project chose 10
large stocks from different sectors and industries to reduce the firm-specific risk. And large
stocks have relatively low standard deviation than small stocks. Following chart is the
information of the 10 large stocks.
Name
Apple Inc.
Starbucks
JPMorgan Chase
General Electric
Ticker
AAPL
SBUX
JPM
GE
BP PLC
Johnson & Johnson
American Electric
Power Co Inc.
Procter & Gamble
BP
JNJ
AEP
Wal-Mart
Microsoft Cor.
WMT
MSFT
PG
Sector
Technology
Services
Financial
Industrial
Goods
Basic Materials
Healthcare
Utilities
Industry
Personal Computers
Specialty Eateries
Money Center Banks
Diversified Machinery
Market Cap
552.67B
37.71B
154.22B
220.00B
Morningstar Category
Large Growth
Large Growth
Large Value
Large Value
Major Integrated Oil & Gas
Drug Manufactures -Major
Electric Utilities
132.22B
190.83B
20.06B
Large Value
Large Value
Large Value
Consumer
Goods
Services
Technology
Personal Products
189.59B
Large Core
Discount, Variety Stores
Application Software
235.14B
230.00B
Large Core
Large Core
The historical price data of the 10 large stocks comes from Yahoo finance. And period of the
historical prices is 11 years, from Jan 1st 2000 to Dec 31st 2011. Next, the project calculated
monthly returns of each stock, and get the average monthly returns, variance, and standard
deviation of the 10 stocks. From the monthly returns and average monthly returns, we got excess
returns, which is crucial in calculating the variance-covariance matrix of the 10 stocks.
Variance-covariance matrix S is the covariance between each two of the 10 stocks. With the
covariance matrix, average returns, and constant c, we can identify envelope portfolios. Using
the formula E(r) – c = S*z, we can get envelope portfolio vector z. From vector z, we could
compute the portion wi of each stocks in envelope portfolios. By setting constant c equals 0 and
0.04, we have our two envelope portfolios x and y. One assumption using this model is that there
is no short sale restriction. Following table is the portions of 10 stocks in the two envelope
portfolios.
AAPL
SBUX
JPM
GE
BP
JNJ
AEP
PG
WMT
MSFT
Sum
Envelope Portfolio X with c=0
27.68%
40.92%
-12.70%
-38.95%
1.45%
36.26%
35.52%
19.82%
16.82%
-26.82%
1
Envelope Portfolio Y with c=0.04
62.04%
90.50%
-24.71%
-89.19%
-6.47%
62.74%
68.81%
10.87%
-0.29%
-74.30%
1
From various combinations of x and y, we found the efficient frontier of the 10 large stocks. The
annual return goes from 5.62% to 43.43%, and standard deviation from 3.32% to 12.8%. Higher
returns have higher risks.
Global Minimum Variance Portfolio (GMVP)
When we got the efficient frontier, we had lots of combinations of the 10 stocks to choose from.
But which portfolio should we invest? The Global Minimum Variance Portfolio (GMVP) model
gives us an option. Literally, GMVP is the combination that has the minimum variance on the
efficient frontier. This portfolio is a good choice for risk-avoid investors.
Given the variance-covariance matrix and average returns, we got the portions of the 10 stocks
below: AAPL 5.17%, SBUX 8.44%, JPM -4.83%, GE -6.04%, BP 6.64%, JNJ 18.91%, AEP
13.72%, PG 25.69%, WMT 28.03% ,MSFT 4.28%. The GMVP has an average annual return
6.62% and standard deviation 3.30%.
Black-Litterman Approach to Optimize Portfolio
We used historical prices to calculate two envelope portfolios, the whole efficient frontier, and
GMVP. It gives investors some clues on portfolio forming. However, it also can be some kind of
naive. The envelope portfolios’ portion table above shows a 38.95% short-sale on GE in
portfolio x, and a negative 89.19% portion of GE in portfolio y. It’s unrealistic in the real world
to have such a large amount short sale in the portfolio. Also, historical returns cannot represent
the returns in the future.
In order to optimize the portfolio, we introduce Black-Litterman approach. The idea of BlackLitterman is that market knows what it is doing and it would be very difficult for investors to
beat the benchmark portfolios. Therefore, the first step of Black-Litterman is to build a
benchmark portfolio using the 10 large stocks. And the benchmark proportion of each stock is
the current market weight. We estimated the expected benchmark return over the next month will
be 1% and the current 3 month t-bill rate is 0.1%. By using var-cov and correlation matrix, the
Black-Litterman model gives us the expected monthly return of each stock and exactly the same
portions as the current market weights.
The second step of Black-Litterman is to add the opinion of analyst. Looking the next month
expected returns of 10 stocks, I made a positive adjustment on APPL, SBUX, JNJ, PG, and
WMT. With the adjusted returns, the model gave a new portion of each stock. Following is the
result:
Optimized
Proportion
AAPL
SBUX
JPM
GE
BP
JNJ
AEP
PG
WMT
MSFT
22.15%
5.85%
5.90%
8.55%
6.43%
15.17%
-0.20%
9.44%
17.48%
9.22%
Gretl Application on Efficient Frontier
Gretl is the second technique I choose to do the efficient frontier. I input the monthly return data
of the 10 stocks, and write commends to calculate average returns, standard deviation, variance-
covariance and correlation matrix, and envelope portfolios. It’s a good check on what we have
done with excel. The output of the gretl is the same with excel.
Results and Conclusion
Through the financial models, we got investment options with the 10 large stocks. Some points
on the efficient frontier have really high returns, and their standard deviations are not so high
compared to portfolios contained small stocks. However, those portfolios are not realistic in the
real word, because they have large amount of short-sale on some stocks.
The GMVP has more reasonable stock weights. And due to its lowest risk, it has a relatively low
annual return 6.62%. The Black-Litterman approach combined market and analyst opinions. The
annual return is 12%, which is an anticipated input. And the portfolio proportions are more
reasonable than some envelope portfolios. However, compared to historical data approach,
Black-Litterman is a little bit subjective. It counts more on the experience and judgment of
investors.
What I learned
From this project, I learned how to make an investment decision based on financial models. I
gained deeper understanding on matrix, envelope portfolio, efficient frontier, and BlackLitterman theory. I learned that only using historical data to predict future performance is naive
and unrealistic on some point. Moreover, I gained a valuable experience on writing gretl
commends and applying them in the investment decision making process.