DIVERSIFICATION AND THE EFFICIENT FRONTIER CONSTRUCTION

In brief, you need to construct the efficient frontier of TWO hedge funds, which you need to select on your own from the data file, and two stock indices, also from the data file. Then need to construct the set of efficient portfolios (i.e. the efficient frontier) of these four risky assets.

Principles of Investment (FIN 351)
Learning Exercise 5
EFFICIENT FRONTIER AND CAPM
PART I – DIVERSIFICATION AND THE EFFICIENT FRONTIER CONSTRUCTION
1. Motivation and Objectives
The main objective of the first part of LE 5 is to learn how to construct an optimal portfolio of
risky securities. A portfolio is optimal if it pays the highest possible return for the level of risk an
investor feels comfortable to take on. Your task is to allocate the available wealth among several
assets to achieve maximum return for each level of risk.
The second objective of this part of LE 5 is to learn about hedge funds. Hedge funds continue to
rise in popularity among investors, yet there is a lot of confusion about them. Hedge funds are
private investment pools, which frequently take concentrated and leveraged positions in illiquid
markets or assets and generally dislike disclosing information about their investment process and
market bets. Hedge fund managers seem to be concerned that public disclosures could wipe out
future expected gains or harden their ability to unwind positions when necessary.
Hedge funds seem to provide an opportunity for reducing the risk of portfolio losses in turbulent
economic times in increasingly globalized financial markets. The globalization of financial
markets brought numerous benefits to investors, but the diversification of risk in integrated
markets became a more challenging task. Returns on international stocks and bonds move more
and more in lockstep with each other, especially in downturns. To overcome this difficulty,
investors allocate larger portions of their wealth to alternative assets, such as hedge funds. You
will investigate the effect of such investments on your portfolio risk and return.
You will work with an interesting data set. It contains returns on hedge fund ind6exes. Each
hedge fund index is a portfolio of several hedge funds, which use similar trading strategies.
Hedge funds are speculative investment vehicles designed to exploit superior information held
by their managers. Information-based trading requires that the information about trading is kept
secret. For this reason, it is very difficult to obtain return data on individual hedge funds.
However, the returns on a hedge fund index and returns on each constituent individual hedge
fund, comprising this index are highly correlated (i.e. very similar).
2. Data
A. The data file Data for LE 5.xlsx (posted on the Blackboard) contains monthly rates of
return (ROR) on ten hedge fund indexes constructed to track the performance of the ten
most popular hedge fund investment strategies.
B. This file also contains monthly returns on the S&P 500 index and the Index of Asia
Pacific stocks.
3. Instructions for Deliverables
Before starting this part of the LE, you need
1. Watch the video and/or study the presentation on Diversification and the Efficient
Frontier
2. Read the Review file for this lecture.
3. Watch the video with the example on portfolio optimization and study the Excel file
with this example.
4. Make sure that Data Analysis and Solver Add-ins are installed in your Excel.
a. Go to File > Options. …
b. Click Add-Ins,
c. In the Manage box, select Excel Add-Ins
d. Click Go.
e. In the Add-Ins available box, select Analysis ToolPack and Solver Add-ins
check boxes, and then click OK.
f. After you load the Analysis ToolPack and Solver Add-ins, the Data Analysis
and Solver commands are available on the Data tab.
g.
You will use Excel for your calculations, but the final report file should be in Word. Please,
submit the final report saved under your last name in Word and the Excel file with
calculations.
5. In your final report, you need to write detailed explanations of the work you
completed to construct the efficient frontier. You report should include the
following:
a. Summary statistics for your data, properly organized. Include the annualized
mean return and standard deviation for each of the four constituent assets,
the variance-covariance matrix (annualized), and the correlation matrix.
b. Description of the optimization problem you solve (with the help of Solver)
to find the weights of the assets in the Minimum Variance Efficient Portfolio
(MVE portfolio). Explain what is special about this portfolio. You need to
explain:
i. The formula for the variance function of your portfolio
ii. What you need to do with the variance function in order to construct
MVE portfolio
iii. Which variables you can choose to achieve your goal
c. Numerical description of the MVE portfolio:
i. Show weights for all constituent assets,
ii. Show the annualized expected return and standard deviation of the
MVE portfolio
d. Description of the mathematical optimization problem, which you solve to
find weights of constituent assets in all other efficient portfolios. Highlight
the difference between the optimization problem you need to solve (again
with the help of Solver) to find the next ten efficient portfolios and the
problem, which you solved to find weights of the MVE portfolio.
e. Numerical description of each of the (at least) 10 efficient portfolios. It is
convenient to organize this information in a table(s). You need to:
i. Report the weight of each asset in each portfolio
ii. Report annualized expected returns and standard deviations for each
of your 10 portfolios
6. Submit the graph of the efficient frontier (copied from the Excel. Clearly mark the
MVE portfolio on this graph.
7. Careful explanation and interpretation of your results IS THE MOST
IMPORTANT REQUIREMENT and will be rewarded with good grades.
4. Deliverables
Deliverable 1:
From the list of 10 hedge fund indices, please choose two indices. You can choose two indices,
which follow strategies that you would like to find out more about.
a. Explain the motivation for each of your two strategies.
b. Explain how each strategy is constructed:
i. Which assets are used?
ii. Which assets are long and which are short?
iii. You can give your own simple example here, using actual stocks or bonds.
c. In what circumstances the strategy makes (loses) money.
Deliverable 2:
1. Using commands in the Excel Functions menu (click on fx button), calculate expected
monthly returns on all assets, i.e. two hedge fund indices and two stock indices (use
AVERAGE function).
2. Annualize your expected returns. It means that you need to multiply monthly returns
by 12. Arrange annualized returns in one row (4 cells).
Deliverable 3:
1. Calculate the variances on all your assets and covariances between each pair. Arrange
them into a variance-covariance matrix.
With four assets in your portfolio, the Variance-Covariance is a 4 by 4 table. On the diagonal of
this matrix are variances, on off-diagonal are covariances, each covariance appears twice since
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐶𝑜𝑣(𝑌, 𝑋).
The steps are:
a. On the Data tab, click Data Analysis.
b. In the Data Analysis window, select Covariance and click OK
c. In the Covariance window enter the Input range, i.e. your data range, check on Labels if
you want them, select the output range. Since your work with four assets, your output
range is 4 by 4 area in Excel or 5 by 5, if you wish to have labels for your variancecovariance matrix. Click OK.
d. You will see the lower triangle of the variance-covariance matrix. Copy covariances from
the lower triangle into the upper triangle.
2. Annualize the variance covariance matrix. It means you have to multiply monthly
variances and covariances by 12. We can do this because we assume markets to be
efficient, at least in a weak form. Market efficiency means that monthly returns are
independent of each other; that is the correlation coefficient between returns in one
month and returns in any other month is 0.
Deliverable 4: Calculate the correlation matrix.
a. Go back to Data Analysis and this time select Correlation and click OK.

b. In the Correlation window enter the data range, check on Labels if you want them,
select the output range, and click OK.
Optimal Portfolio Selection
You are ready to learn to use mean-variance optimization framework to construct the efficient
frontier. The efficient frontier is the set of the optimal portfolios. Each portfolio on the frontier is
optimal in a sense that it provides the highest possible expected return per unit of risk. You need
to use the Solver optimization package in Excel to complete this part.
Deliverable 5: Construct the Minimum Variance Efficient Portfolio of four risky assets.
Step 1: Open the Excel spreadsheet containing annualized expected returns on your assets and
the annualized variance-covariance matrix.
Below, enter starting weights for all assets. You can start with any set of numbers for weights.
They will change after you run Solver.
In the next column, enter a sum of weights formula (use =SUM( ) command in Excel). We will
use this cell to set up a constraint:
∑𝑤𝑖 = 1
𝑁
𝑖=1
This constraint simply states that weights of four assets in your portfolio should sum to one.
Step 2: In the next three columns enter formulas for the variance, standard deviation and
expected return (in that order) of a portfolio:
𝜎𝑝𝑓
2 = ∑ 𝑤𝑖
2𝜎𝑖
4 2
𝑖=1 + ∑ ∑𝑖≠𝑗 𝑤𝑖𝑤𝑗𝜎𝑖𝑗
4
𝑗=1
(1)
𝜎𝑝𝑓 = √𝜎𝑝𝑓
2
(2)
𝐸[𝑟𝑝𝑓] = ∑ 𝑤𝑖𝐸[𝑟𝑖
]
4
𝑖=1
(3)
The more efficient way to calculate variance of a portfolio and its expected return is to use
matrices and matrix multiplication commands in Excel.
We will use the following matrix form for equation (1):
𝜎𝑝𝑓
2 = 𝜔Ω𝜔′ (4)
For the portfolio of four assets, in equation (4),  denotes 1 by 4 row vector of weights,  stands
for 4 by 4 variance-covariance matrix, and ’ is a transpose of , i.e. a 4 by 1 column vector of
weights.
Use the following command in Excel to perform matrix multiplication to find the variance of
your first portfolio
=MMULT(MMULT(1 by 4 range of cells for the weight vector , 4 by 4 range of
cells for var-covar matrix ),TRANSPOSE(1 by 4 range of cells for the weight
vector ))
Press simultaneously CTRL-SHIFT and, while holding these two keys, also press ENTER.
In the next cell, calculate the standard deviation of your portfolio, which is just a square root of
the variance
Step 3: Calculate the expected return on your portfolio. In the matrix notation, equation (2)
becomes:
𝐸[𝑟𝑝𝑓] = 𝐸𝜔′ (5)
In equation (5)  stands for the 1 by 4 row vector of annualized expected returns and ’ is a
column 4 by 1 vector of weights.
To do this in Excel, write:
=MMULT(1 by 4 range of cells for the vector of the expected returns E,
TRANSPOSE(1 by 4 range of cells for the vector of weights ))
Do not forget to press simultaneously CTRL-SHIFT and while holding these two keys, also press
ENTER.
When you write the Excel version of the variance formula (4) and expected return formula (5),
“lock” the addresses of cells, which contain the variance-covariance matrix and the vector of
expected returns. Do not lock addresses of cells, which contain weights1
.
Step 4: Use Solver to find the weights of the minimum variance efficient (MVE) portfolio. This
is a portfolio with the lowest possible variance and your objective here is to minimize the
variance without any consideration for the portfolio return (i.e. no constraints on the expected
return of the portfolio needed).
a. Go to Data and click on Solver
b. In Solver window, in “Set Objective”, enter the address of the cell
containing the variance function of the MVE portfolio, then click on Min.
Your objective is to minimize the variance of the portfolio.

1 Copy everything 10 times (next ten rows). Notice that addresses of the cells where you have
weights change, but the cells for variance-covariance matrix and the vector of expected returns
remain the same.
c. In “By Changing Cells” enter the range of weights that Solver can change. Make
sure that you do not restrict short sales, i.e. don’t check the box for positive
values. Negative weights simply mean that you have a short position.
d. Add the constraint on weights. In the Constraint window, in the reference cell
place, enter the address of the cell with the sum of weights formula and set it to
equal to 1:

e. Click OK. The weight constraint appears in the constraint window.
f. Now click on Solve
Deliverable 6:
Use Solver to calculate efficient portfolios for higher expected returns. You need to increase
expected returns above the level of E[RMVE] in small increments (for example, the next
portfolio’s expected return would be 1% higher than the expected return on the MVE portfolio).
The expected return on the third portfolio would be 1% higher than the expected return on the
second portfolio and 2% higher than the expected return on the MVE portfolio, etc. Now, the
expected return constraint matters. You need to construct a portfolio, which pays your desired
expected return with the smallest possible variance (and standard deviation). Mathematically,
you solve the following optimization problem:
min
𝑤1,𝑤2…..𝑤𝑁
𝜎𝑝𝑓
2
(6)
subject to the following constraints:
∑𝑤𝑖 = 1
𝑁
𝑖=1
𝐸[𝑟𝑝𝑓] = 𝐴,
where A is your chosen expected return.
You will need to repeat your optimization with Solver 10 more times, each time increasing
the expected return by 1%. Don’t use paste and copy at this point.
In Solver, the steps are the same as in Step 4, except that in the “Subject to constraints”
window, you now need to add the return constraint.
a. Click on “Add”. The constraint window will open.
b. In the “Cell Reference”, enter the address of the cell, which contains the expected
return on the second portfolio after the MVE portfolio, and choose =.
c. In the “Constraint” enter = address of cell containing the expected return on the
MVE pf and add 1% to it: = 𝐸[𝑟𝑀𝑉𝐸] + 0.01. This means that return on your next
portfolio should be equal to the return on MVE portfolio plus 1%.
d. Click on “OK.”
Appendix: List of Web Resources on Hedge Funds
1. Investopedia is always a good place to start:
http://www.investopedia.com/university/hedge-fund/strategies.asp
2. Concise description of hedge fund strategies:
http://www.magnum.com/hedgefunds/strategies.asp
3. An excellent and non-technical article from Wall Street Mojo website that describes
several of the most popular strategies and provides simple examples for each of them.

4. More formal “Primer on Hedge Funds” by William Fung and David Hsieh
http://www.starkresearch.com/services/documents/hedge_funds/HFprimer.pdf
5. An excellent but more technical article “Hedge Funds: Risk and Return” by Burton G.
Malkiel and Atanu Saha
http://www.cfapubs.org/doi/pdf/10.2469/faj.v61.n6.2775
6. “How Do Hedge Funds Get Away with It? Eight Theories” by John Cassidy, May 14,
2014
http://www.newyorker.com/news/john-cassidy/how-do-hedge-funds-get-away-with-iteight-theories
PART II – THEORETICAL PROBLEMS
1. (a) The risk-free rate of return is 8%, the required rate of return on the market,
𝑬[𝒓𝒎] is 12%, and Stock X has a beta coefficient of 1.4. If the dividend
expected during the coming year, D1, is $2.50 and the constant growth rate,
𝒈 = 𝟓%, at what price should Stock X sell?
(b) Now suppose the following events occur:
(1) The Federal Reserve Board increases the money supply, causing the riskless
rate to drop to 7%.
(2) Investors’ risk aversion declines: this fact, combined with the decline in 𝒓𝒇
,
causes 𝑬[𝒓𝒎] to fall to 10%.
(3) Firm X has a change in management. The new group institutes policies that
increase the growth rate to 6%. Also, the new management stabilizes sales
and profits, and thus causes the beta coefficient to decline from 1.4 to 1.1.
After all these changes, what is Stock X’s new equilibrium price? (Note: D1 goes
to $2.52.)
2. (a) Suppose Carter Chemical Company’s management conducts a study and concludes
that if Carter expands its consumer products division (which is less risky than its
primary business, industrial chemicals), the firm’s beta will decline from 1.1 to 0.9.
However, consumer products have a somewhat lower profit margin, and this will
cause Carter’s growth rate in earnings and dividends to fall from 7 percent to 6
percent. Should management make the change? Assume the following:
𝑬[𝒓𝒎] = 𝟏𝟎%; 𝒓𝒇 = 𝟕. 𝟓%; 𝑫𝟎 = $𝟐.
(b) Assume all the facts as given in part (a), except the one about the changing beta
coefficient. By how much would the beta have to decline to cause the expansion to be
a good one? (Hint: set P0 under the new policy equal to P0 under the old one, and
find the new beta that produces this equality.)
.
3. You are holding a portfolio of stocks where the beta of your portfolio is 2 and its
correlation with the market portfolio, M, is 0.5. The risk free rate is 5%, the expected
market return is 10%, and the standard deviation of the market return is 15%. How
much risk reduction could you achieve, at no sacrifice in expected return, by making
your portfolio an efficient one?
4. Given that the risk-free rate is 10%, the expected return on the market
portfolio is 20%, and the standard deviation of returns to the market
portfolio is 20%, answer the following questions:
a. What is the slope of the capital market line?
b. You have $100,000 to invest. How should you allocate your
wealth among risk free assets and the market portfolio in
order to have a 25% expected return?
c. What is the standard deviation of your portfolio in b)?
d. What is the correlation between the portfolio in b) and the
market portfolio?
e. Suppose that the market pays either 40% or 0% each with
probability one half. You alter your portfolio to a more risky
level by borrowing $50,000 and investing it and your own
$100,000 in M. Give the probability distribution of your wealth
(in dollars) next period.
5. Consider two mutual funds, A and B. The beta for fund A is 0.60, and the standard
deviation of the rate of return = 0.20. The beta for fund B. is 1.30 and its standard
deviation of the rate of return = 0.325. The standard deviation of the market portfolio
is 0.25. Are these funds as well diversified as possible?
6. You have been provided the following data on the securities of three firms and the
market:
Security 𝑬[𝒓𝒋
] 𝝈𝒋 𝝆𝒋𝒎 𝜷𝒋
Firm A 0.13 0.12 ? 0.9
Firm B 0.16 ? 0.40 1.10
Firm C 0.25 0.24 0.75 ?
Market 0.15 0.10 ? ?
Risk-free 0.05 ? ? ?
Assume the CAPM holds true.
a. Fill in the missing values in the table.
b. What is your investment recommendation on each asset? Buy or sell?
c. Suppose that you are currently holding a portfolio consisting of Firm B
only. If you increase your portfolio weight on Firm B by 0.2 (or 20%)
and borrow the needed money at the risk-free rate, what will be the
new standard deviation of your portfolio?

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