This document describes a portfolio optimization project. It analyzes historical stock return data for Apple and Netflix to construct an efficient frontier. A risk-free rate is calculated from treasury bill returns. An optimal risky portfolio is determined by maximizing the Sharpe ratio. Based on a risk aversion index of 1, the appropriate weights in the optimal portfolio and risk-free asset are calculated to maximize utility. Graphs of the efficient frontier, capital allocation line, and indifference curve illustrate the optimal portfolio selection.

1. Project 1
Fnu Suolongfu
UNIVERSITY OF MINNESOTA
Portfolio Theory
 Part I:
Raw Score = 64
Sensation-seekers thrive on new, intense, and varied situations. Their personalities are
associated with risk-taking because sensation-seeking drives individuals to seek out
highly stimulating experiences, which often include risk. Sensation-seekers have strong
positive reactions to intense stimuli. While there are many constructive aspects of this
personality type, those with this trait often take more risks, are more impulsive, and
become bored more easily. In certain ways, a sensation-seeking personality is an asset
- such individuals thrive on stress, action, uncertainty, and challenge. In other ways, it
is a liability - they may take outlandish risks. Low sensation-seekers, on the other hand,
are reliable, can handle monotony, and prefer to sleep on their decisions. They avoid
novel and stimulating experiences.
According to your results, you are a sensation seeker. Sensation-seeking can take the
form of searching out harmless yet invigorating stimuli such as art or music, or traveling
to an exotic locale. It can also refer to more dangerous risks, intended to achieve an
adrenaline rush. You seek out new experiences, and may become bored by repetitive,
routine tasks. While you enjoy the thrill of risk-taking, your behavior is rarely extreme
or reckless.
According to the table below, my risk aversion index is 1.
Raw Score Range Risk Aversion Parameter
0-20 2.5
20-40 2
40-60 1.5
60-80 1
80-100 0.5
 Part II:
 The components of my risky portfolio contains two different stocks:
Stock a = Apple Inc.

2. Stock b = Netflix Inc.
 The Graph of the monthly stock return between 1/2005 and 11/2015 for Apple Inc.:
 The Graph of the monthly stock return between 1/2005 and 11/2015 for Netflix
Inc.:
Firstly, I retrieved data of historical prices from finance Yahoo, then I calculated every
monthly stock return by using the current month Adjusted Closing Price minus last
month Adjusted Closing Price get the change in Adjusted Closing Price, then divide
this change in Adjusted Closing Price by the last month Adjusted Closing Price to get
the monthly stock return. And draw the graph like above two.
-40.00%
-30.00%
-20.00%
-10.00%
0.00%
10.00%
20.00%
30.00%
1/14/2004 10/10/2006 7/6/2009 4/1/2012 12/27/2014 9/22/2017
Apple Inc
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
1/14/20045/28/200510/10/20062/22/20087/6/200911/18/20104/1/20128/14/201312/27/20145/10/20169/22/2017
Netflix Inc

3.  The Summary Statistics on each stock
Mean: Calculated from the sum of each monthly return divided by number of period,
which is the expected return over the time period.
Variance: Calculated from the Means and monthly return by using the VARP function.
Skewness: Calculated by the SKEW function excel.
Kurtosis: Calculated by the KURT function in excel.
Median: Got by the Median function in excel.
Inter-quartile range: Calculated by the QUARTILE function in excel.
Std Devn: Calculated by the positive square root of Variance.
Covariance and Correlation: Calculated by the COVAR and CORREL functions in
excel.
The Portfolio statistics are given in column F with holding 50% of Apple Inc. stocks
and 50% of Netflix Inc. stocks.
rapple = 0.0293, rnetflix = 0.0467, 𝜎apple=0.09554, 𝜎Netflix=0.169716
Portfolio return of two stocks:
E(rp)=wappleE(rapple)+wnetflixE(rnetflix)
Portfolio variance of two stocks:
𝜎p
2=𝑤apple
2 𝜎apple
2+wnetflix
2 𝜎netflix
2+2wapple 𝑤Netflix*Cov(rapple, rnetflix)
As we can see that the mean (expected return) of Apple Inc. is smaller than mean
(expected return) of Netflix Inc, but the Standard deviation of Apple Inc is smaller than

4. Netflix Inc., which means Netflix Inc. stocks are riskier than the stocks of Apple Inc.,
but with a higher return. So we will then obtain the risk free return in order to get all
the data we want to find the optimal portfolio.
 Part III:
 The Graph of the returns to one-month treasury bills for the past 10 years.
I retrieved one-month “Treasury Bill: Secondary Market Rate” data for the past 10 years
from the Federal Reserve Bank of St. Louis in which we can see that the monthly return
of T-bill has a large increase over the financial crisis period and then goes lower.
 The Summary Statistics for the returns to one-month treasury bills for the past 10
years.
0
0.01
0.02
0.03
0.04
0.05
0.06
2004-01-142005-05-282006-10-102008-02-222009-07-062010-11-182012-04-012013-08-142014-12-272016-05-10
Treasury Bill: Secondary MarketRate

5. Mean: Calculated from the sum of each monthly return divided by number of period,
which is the expected return over the time period.
Variance: Calculated from the Means and monthly return by using the VARP function.
Skewness: Calculated by the SKEW function excel.
Median: Got by the Median function in excel.
Std Devn: Calculated by the positive square root of Variance.
 The Expect Return of T-bills over this time period is 1.3%. So our best estimate of
the risk-free rate is the expected return to one-month Treasury bills over the past
ten years. So our rf = 0.013.
 Part IV
 We have data. For Apple Inc. stocks, Netflix Inc. stocks and T-bills (risk-free assets)
rapple = 0.0293, rnetflix = 0.0467, 𝜎apple=0.09554, 𝜎Netflix=0.169716
Cov(rapple, rnetflix)= 0.001176
Portfolio return of two stocks:
E(rp)=wappleE(rapple)+wnetflixE(rnetflix)
Portfolio variance of two stocks:
𝜎p
2=𝑤apple
2 𝜎apple
2+wnetflix
2 𝜎netflix
2+2wapple 𝑤Netflix*Cov(rapple, rnetflix)
 Construct efficient frontier: for each level of expectation, find optimal combination
of weights to minimize variance

6. The column L is the weight in Apple Inc.
The column M is the weight in Netflix Inc.
Then we use E(rp)=wappleE(rapple)+wnetflixE(rnetflix) to calculate the expectedrisky portfolio
return for each combination of Apple Inc. stock and Netflix Inc. stock. Show in column
N.
Then we use 𝜎p
2=𝑤apple
2 𝜎apple
2+wnetflix
2 𝜎netflix
2+2wapple 𝑤Netflix*Cov (rapple, rnetflix) to
calculate the risky portfolio variance for each combination of Apple Inc. stock and
Netflix Inc. stock, show in column O. And the risky portfolio standard deviation is
just the square root of the risky portfolio variance, show in column P
The Risk Premium is calculated the difference between the expected risky portfolio
return minus risk free rate of return. Which equals to E(rp) – rf, rf = 0.013 in our case.
Then we can use the data from excel to get the minimum variance frontier：
X-axis denotes the standard deviation. We use the data in column P.
Y-axis denotes the expected return. We use the data in column N.
As we can see in the graph, this is actually the Minimum Variance Frontier, the actual
Efficient Frontier is the part of minimum variance frontier eliminating the dominated
Portfolio.
All the points being labeled on the Efficient Frontier are the Opportunity Set of Risky
Assets
4.666%
4.493%
4.319%
4.146%
3.973%
3.800%
3.627%
3.453%
3.280%
3.107%
2.934%
0.000%
0.500%
1.000%
1.500%
2.000%
2.500%
3.000%
3.500%
4.000%
4.500%
5.000%
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Efficient Frontier

7.  Optimal Risky Portfolio:
Sharpe Ratio is the Risk Premium E(rp) – rf divided by the standard portfolio
deviation 𝜎p.
For each point in the Portfolio Opportunity Set, we can draw a CAL from the risk-free
point.
As the graph below shows, our goal is to push the CAL as far to the left as possible,
which increases expected return while decreasing risk.
That is, we wish to maximize the Sharpe Ratio, Sp = (E(rp) – rf )/ 𝜎p, subject to
Wapple+Wnetflix=1, By choosing Wapple to get the Optimal Risky Portfolio.
I use Solver in the excel to solve this maximization problem, simply choose Max Sharp
Ratio, Sp by changing weight of Apple Inc. stock, Wapple .
Finally, we get the result below:
4.666%4.493%4.319%4.146%3.973%3.800%3.627%3.453%3.280%3.107%2.934%
0.0130
0.000%
1.000%
2.000%
3.000%
4.000%
5.000%
6.000%
7.000%
0 0.05 0.1 0.15 0.2 0.25
Efficient Frontier
CAL

8. The P280 cell in the graph above is the optimal weight we ought to put on Apple Inc.
stock Wapple = 0.599 and the P281 cell in the graph above is the optimal weight we
ought to put on Netflix Inc. stock Wnetflix = 0.401 to form our optimal risky portfolio.
The Expected Return of optimal portfolio is 0.03628.
The Variance of optimal portfolio is 0.008471271.
 Part V:
 The appropriate weights of the risky portfolio and the riskless T-bills in our
overall optimal portfolio
According to Markowitz Theorem, every investor, regardless of risk aversion should
choose the same risky portfolio. This result in the optimal CAL what we already get
from Part IV. The investors with different risk-aversion will then choose different
mixtures of the risk-free asset and the optimal risky portfolio. For me, My Risk
Aversion Index is 1
So what we need to do in this part is Maximize Utility: U=C=E(r) – (1/2)*Avar(r)
From E(r)=yE(rp)+(1-y)rf and Risk Premium Rp=E(rp) – rf, We can get that U = rf +
yRp – (1/2)y2Var(r), and Var(r) is just the Variance of optimal portfolio. So I use
solver in excel again to maximize Utility by choosing y, which is the appropriate
weights of the risky portfolio in our overall optimal portfolio.
Then we get the appropriate weights of the risky portfolio and the riskless T-bills in
our overall optimal portfolio are 2.74 and -1.74 respectively.

9. In this situation, we are doing short sale of the risky portfolio because y greater than 1.
 Graph the capital allocation line and the tangential indifference curve to optimal
portfolio.
What we need to do in this step is to graph an indifference curve and add it into the
last optimal CAL and Efficient Frontier graph.
For graphing the indifference curve, we already get the maximized utility which is
C= 0.049, if X-axis is standard deviation and Y-axis is expect return, We graph
Y = C+(1/2)AX2 = 0.049+(1/2)X2 as the graph above.
 The percentage breakdown of the three individual components of the overall
portfolio, meaning of each of the two stocks and the T-bills.
The percentage of the T-bills of the overall portfolio = -174%
The percentage of the Apple Inc stock of the overall portfolio = 2.74* 59.9% =
164.126%
The percentage of the Netflix Inc stock of the overall portfolio = 2.74* 40.1% =
109.874%
4.666%4.493%4.319%4.146%3.973%3.800%3.627%3.453%3.280%3.107%2.934%
0.0130
0.000%
2.000%
4.000%
6.000%
8.000%
10.000%
12.000%
14.000%
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Efficient Frontier risk free return CAL Indifference Curve

10.  Part IV: Summary
From this experiment, I learned how to collect data and how to use statistical software
like Excel and R to get expectation, variance, covariance and different significant
statistical summaries to identify the risk-return combinations available from the set of
risky assets. I learned how to draw the minimum-variance frontier by using the
opportunity set of risky assets and also how can we get efficient frontier from
minimum-variance frontier. By using Excel, I learned that how to use graph function
of Excel to draw different kind of lines like CAL, Indifference curve and how to use
functions in Excel. From Excel, I also learned a strong tool called solver for
maximization and minimization problem, for example, maximize Sharpe ratio and
Utility in this project.
I personally experienced how to use Markowitz Portfolio Selection to get the optimal
risky asset and then the allocation to risk-free asset and risky portfolio for different
investor with different risk aversion index. In the Part I, we adequately interpret the
meaning of risk aversion index and how it is decided.
This Project helps me fully understand how to analysis the optimal portfolio selection
practically using the real data and how to write a report like this.