The document discusses the author's analysis of various stock portfolios from February 2015 to December 2015. For each month, the author calculates an equal weighted portfolio, global minimum variance portfolio, targeted return portfolio, and tangency portfolio using stock data from Apple, Exxon Mobil, Microsoft, and Alphabet. The author then analyzes the expected returns and risks of each portfolio to determine which would provide the highest utility for a risk-seeking investor. The optimal portfolio choice depends on the monthly returns and whether a tangency portfolio can be computed given the risk-free rate.

1. Ismael Correa
Econ 490 | Financial Econometrics: Final Report
May 15th, 2016
The Tangency Portfolio
Apple, Inc. (AAPL)
Exxon Mobil Corporation (XOM)
Microsoft (MSFT)
Alphabet, Inc. (GOOGL)
Alphabet, Inc. (GOOG)
In order to leverage the concepts that we learned on portfolio theory, I decided to
focus my final report on creating multiple portfolios, including an equal weighted portfolio,
global minimum variance portfolio, targeted return portfolio and a tangency portfolio. The
main question I wanted to ask was what was the tangency portfolio for each month from
February 2015 until December 2015. This tangency portfolio would give me the best
Sharpe ratio that could be located within a portfolio and effectively optimize my utility as a
risk-seeking investor.
The stocks that I examined were Apple Inc. (AAPL) Exxon Mobil Corporation (XOM),
Microsoft (MSFT), Alphabet Inc. (GOOGL), and Alphabet Inc. (GOOG). I leveraged both excel
and R in my analysis through extracting the data from yahoo finance into excel in order to
have all the information I needed I needed to utilize these adjusted prices to compute the
data that I needed. From these adjusted prices, I found their expected return by taking the
holding period return of each stock, including dividends, from one month to the next. Then
I created a 5 x 5 covariance matrix, which consisted of the variance of each stock followed
by the covariance of each stock with the other. After computing this information I found the
beta of each stock from the year of 2015 by taking the covariance of each stock with the
Standard and Poor’s 500, and diving this number by the variance of the Standard and
Poor’s 500. This information would later on be used to compute the alpha of each stock by
taking the return of the portfolio minus the risk free rate plus the return on the market
minus the risk free rate multiplied by the beta.
After getting all of this information, I was able to compute the portfolios for the
months of February 2015 until December 2015. I followed a standard operating procedure
for each month. I started by entering the expected return in a 1 x 5 matrix into R. Followed
by entering the covariance matrix in a 5 x 5 matrix. After this I set the risk free rate to a
hypothetical .005%. I then found the equal weighted portfolio and got the return on this
portfolio as well as the risk associated with it and the shares to be invested in each stock,
which would be .2 in this situation. After finding the equal weighted portfolio, I found the
global minimum of the portfolio that gave me this expected return, the risk associated with
it and the amount of shares to invest in each stock, which often times was leveraged. I then
found the mean-variance efficient portfolio that was set to optimize a portfolio achieved at
a target expected return. The last portfolio I found was the tangency portfolio, which was
my main interest, this portfolio gave me the optimal level of the return and risk for each
month and also gave me the most utility. After compiling these different portfolios I finally
found the efficient frontier and plotted this on a graph along with the global minimum
portfolio, the tangency portfolio and the Sharpe ratio. The optimal amount of shares to
invest in each stock depended on the level of risk I was willing to take on and in this case I
am a risk-seeking investor.

2. Empirical Analysis (February 2015 through December 2015)
ZOOM IN TO SEE THE NUMBERS
February 2015
March 2015
April 2015
May 2015
June 2015
July 2015
August 2015
September 2015

3. October 2015
November 2015
December 2015
Results Analyzed
Each set of portfolios belongs to one of the eleven months analyzed. The portfolios
displayed in the results above consist of an equal weighted portfolio, a global minimum

4. variance portfolio, a target mean-variance portfolio and a tangency portfolio. In addition to
this, some months are accompanied by an efficient frontier if the risk free rate is less than
the average return on the global minimum variance portfolio. If the risk free rate is greater
than average return on the global minimum variance portfolio, then the tangency portfolio
cannot be computed and therefore I could not proceed in finding the optimal return and
risk for the portfolio.
The month of February is the first month in which I will analyze. The equal weighted
portfolio saw an expected portfolio return of 0.06520731 and a portfolio standard
deviation of 35.27706. Each stock was invested at an equal amount of 20%. The global
minimum portfolio saw an expected portfolio return at 0.00903047 and a portfolio risk of
1.741303. This portfolio was leveraged in the sense that some stocks were shorted while
others were invested at an amount greater than one however, the sum of the short
positions and long positions still equaled to one. The efficient portfolio with a target return
equal to the return of Apple for the month of February gave a portfolio expected return of
.1049036 and a portfolio standard deviation of 2.965762, which was a leveraged portfolio
as well. The tangency portfolio gave me the highest portfolio expected return that was
.9782703 and a portfolio standard deviation of 24.53341, this portfolio was also a
leveraged one. In my decision on which portfolio to select, my decision is on the risk-
seeking side as I stated above. The tangency portfolio would thus give me the most utility
since I am able to trade off a higher risk for a higher expected return than we see in the
other portfolios.
The month of March however saw different results. This month had a negative equal
weighted portfolio with an expected portfolio return at -0.03000318 and a portfolio
standard deviation at 35.27706. The global minimum variance portfolio also saw a negative
return that was at -0.01780928 and a portfolio standard deviation at 1.741303. In this
situation, a tangency portfolio cannot be created due to the fact that the risk free rate will
never be higher than the return on the global minimum variance portfolio. The package,
which I was using, did not allow this to occur. Therefore, for this month, even though I am a
risk seeking individual I still would like to limit my losses at the least amount of risk and
therefore, I decide to invest in the risk averse global minimum variance portfolio.
The month of April did not see a negative global minimum variance portfolio.
Instead it was a month that had modest returns on the portfolios. The equal weighted
portfolio had a portfolio expected return of 0.03995662 and a portfolio standard deviation
of 35.27706. The global minimum portfolio was leveraged with a portfolio expected return
of 0.05820585 and a portfolio standard deviation of 1.741303. The target mean-variance
portfolio was targeted at the return of Microsoft and had a return of .1962616 with a
slightly higher portfolio standard deviation of 2.54386 and was also leveraged. The
tangency portfolio had a very high portfolio expected return at .3740369 and an even
higher risk than all three of the other portfolios at 4.585952. Since I am a risk-seeking
investor and love to take my chances, I take on all the risk and go with the tangency
portfolio with the highest expected return than all the other portfolios.
The month of May saw a similar story that the month of March had. The global
minimum variance portfolio had negative returns at -0.0288928 and a portfolio standard
deviation at 1.741303. The equal weighted portfolio had a positive expected return at
0.0003493088 but with an astoundingly high risk for such a small return at 35.27706. The
targeted portfolio of Alphabet, Inc. (GOOGL) saw an expected return of -0.00628681 and a

5. risk of 2.238932. My optimal decision is to invest in the target portfolio since it gives me
the least losses with a moderate level of risk.
The months of June through September all saw negative global minimum variance
portfolios. The best strategy, in my opinion, for the month of June would be to invest in the
target portfolio targeted at Alphabet, Inc. (GOOGL) due to the fact that this portfolio would
give me the least amount of losses when comparing it to the equal weighted portfolio and
the global minimum variance portfolio. For the month of July, the optimal decision is to go
with the targeted portfolio as well due to the fact that this portfolio gives me a way much
higher return of 0.2175025 with a risk of 6.631801. Even though the risk is higher, it would
be irrational to invest in a portfolio that would generate a negative portfolio expected
return or less returns. For the month of August, it would be optimal to invest in the
targeted portfolio aimed at Alphabet, Inc. (GOOGL) which would be the portfolio with the
least amount of losses possible. This is the safest portfolio option that would guard me
from a higher expected loss if I were to invest in the equal weighted portfolio or the global
minimum variance portfolio. For the month of September, it would be optimal to invest in
the targeted portfolio at Microsoft due to the fact that it gives me positive expected returns.
The month of October was a month if moderate returns. The equal weighted
portfolio saw an expected return of 0.1417983 and the portfolio risk was at 35.27706. On
the other hand the global minimum variance portfolio saw an expected return of .1254743
and a portfolio risk of 1.741303. The targeted portfolio had the returns of Microsoft as the
target return and saw a portfolio expected return of 0.1893358 and a portfolio risk of
2.553405. The tangency portfolio had the highest return with 0.1549041 and a portfolio
risk of 1.942378. The efficient frontier was plotted for the month of portfolio. After
considering the investment strategy, as a risk-seeking investor, my most utility would come
from investing in the riskiest portfolio since it would have the most expected return.
However, for the month of October the most expected return came from the tangency
portfolio and this portfolio did not have a high risk. For the month of October, investing in
the tangency portfolio would give me the most utility.
The month of November was a month of trivial returns, with the exception of the
tangency portfolio. The equal weighted portfolio saw an expected return of 0.02577046
and a portfolio risk of 35.27706. The global minimum variance portfolio saw an expected
return of 0.006591484 and a portfolio risk of 1.741303. The target portfolio was targeted
at Microsoft and saw an expected return of 0.04637845 and a portfolio risk of 2.734432.
The tangency portfolio had the highest returns with an expected return of 0.6851032 and a
portfolio standard deviation of 35.99652. As a risk-seeking investor, this portfolio selection
is inclining my investment strategy to lean toward the riskiest portfolio, the tangency
portfolio. Even though it has the highest risk, it still has the highest return by a large
margin.
The month of December was also a month similar to the months of March and May
through September. This month’s investment strategy would be to go toward the target
portfolio aimed at Alphabet, Inc. (GOOG) since this portfolio gives me a positive expected
return when compared to the global minimum variance portfolio and the equal weighted
portfolio.
The set of concepts from portfolio theory allowed me to derive these portfolios and
make my optimal decision based on the utility I get from my risk-seeking investment
strategy. Economically speaking, these portfolios are optimal based on my preferences.

6. Works Cited
1. Zivot, E. (November 11th, 2008) Computing Efficient Portfolios in R.
University of Washington.