Investment Theory and Applications

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1. The entire examination is to be submitted electronically. If there is more than one
file, please put them in a ZIP file and send directly to the TA’s email address (if
you send it to me as well, I will record your submission and archive it).
2. The examination work is due on or before 5 pm Friday December 20, 2013.
3. Each problem is worth 20 points.
4. Show all your work if you want partial credit.
5. Do not do a data dump. To get full credit you need to explain the results you
obtain and summarize your results neatly in tables or use charts to make you
point.
You will be analyzing a collection of equity indices. These indices are building blocks
for several portfolios.
The dataset you will use consists of the monthly total returns of these indices. These data
are found in the file TakeHomeFall2013Data.xls found in the ZIP file.
You can use Excel, R or any other software tool to compute the results in this
examination.

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You will be asked to perform your analysis over the six-year time period from January
31, 2003 to December 31, 2008. The six month period from January 31, 2009 to June 20,
2009 are reserved to evaluate the ex-post performance of your portfolios.
Investment Theory and Applications Dr. Frederick Novomestky
FRE 6711 – Fall 2013 Final Examination
2
Problem 1 (20 points)
The worksheet with tab Equity Market Returns contains the time series of monthly
total returns of the global equity markets and their various segments. The worksheet with
tab Risk Free Returns contains corresponding time series of one month Treasury bill
returns. Prepare a table summarizing the ex post Sharpe ratios for the following
segments: World, Europe, Asia and the U.S. Comment on the ranking of these markets
by Sharpe ratio. What is the unbiased estimate of the standard deviation (volatility) of
the returns for these segments?
The next step is to analyze the performance of Europe, Asia and U.S. using a single
factor model. The single factor model is the return of the World index. Estimate a model
for the each of the following segments using linear regression. The dependent variable is
the segment return in excess of the risk free rate.
 Europe
 Europe Large Cap
 Europe Small Cap
 Europe Value
 Europe Growth
 Asia
 Asia Large Cap
 Asia Small Cap
 Asia Value
 Asia Growth
 U.S.
 U.S. Large Cap
 U.S. Small Cap
 U.S. Value
 U.S. Growth

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Do you observe a significant intercept parameter (alpha) in any of these segments?
Which market segment has the highest alpha and which has the lowest?
Finally, estimate a three factor model consisting of the following factors
 Market Factor – the World index minus the risk free return
 Size Factor – the World Small Cap minus the World Large Cap (a size factor)
 Valuation Factor – the World Va
lue minus World Growth (a valuation factor)
Estimate the three factor model for each of the above segments also using linear
regression.
Do you observe a significant alpha from the three factor models? Which market segment
has the highest alpha and which has the lowest? Is this observation consistent with the
single factor model?
Investment Theory and Applications Dr. Frederick Novomestky
FRE 6711 – Fall 2013 Final Examination
3
Which is a better model: a one factor model or a three factor model? Make your case
using analysis of variance (ANOVA). Does the answer depend on the segment?
Problem 2 (20 points)
Estimate the variance covariance matrix of returns for the following equity market
segments.
 Europe Large Cap
 Europe Small Cap
 Europe Value
 Europe Growth
 Asia Large Cap
 Asia Small Cap
 Asia Value
 Asia Growth
 U.S. Large Cap
 U.S. Small Cap
 U.S. Value
 U.S. Growth
What are the estimated volatilities of these assets (i.e. standard deviation of returns)?
Using these volatilities and the above matrix, construct the matrix of correlation
coefficients. Comment on the correlations of the equity segment indices. Do you
observe blocks of highly correlated segments? What is the estimated volatility of the
World Index?
Investment Theory and Applications Dr. Frederick Novomestky
FRE 6711 – Fall 2013 Final Examination
4
Problem 3 (20 points)
The market sizes for the equity segments as of December 31, 2008 are found in the
worksheet tab Equity Market Sizes and are summarized in the following table.
Asset Class Size as of 12/31/08
EUROPE $ 5,273,160,422,702
EUROPE VALUE $ 2,562,380,167,088
EUROPE GROWTH $ 2,710,780,255,614
EUROPE LARGE CAP $ 4,554,877,685,111
EUROPE SMALL CAP $ 452,577,186,219
ASIA $ 3,172,681,731,520
ASIA VALUE $ 1,584,702,142,681
ASIA GROWTH $ 1,587,979,588,839
ASIA LARGE CAP $ 2,586,744,297,555
ASIA SMALL CAP $ 495,129,995,543
U.S. $ 9,799,521,519,870
U.S. VALUE $ 4,794,705,636,699
U.S. GROWTH $ 5,004,815,883,171
U.S. LARGE CAP $ 8,877,357,459,700
U.S. SMALL CAP $ 747,242,781,386
In this problem you will construct market implied forecasts of equity segment excess
returns (i.e. expected returns in excess of the risk free return).
 Compute the market size weights of the equity segments in Problem 2 (Hint: combine
the market sizes for Europe Asia and the U.S. Derive the segment weights as a
proportion of these total sizes).

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 Compute the risk aversion parameter  as the Sharpe ratio of the World Index from
Problem 1 divided by the estimated volatility of the World Index from Problem 2.
 Derive the market implied expected excess returns from s s   w where s  is the
variance-covariance matrix of segment returns from Problem 2 and s w is the vector
of market size weights
 Derive the market implied expected nominal returns from the market implied
expected excess returns and the risk free returns for the month ending January 31,
2009.
Summarize your results in two sets of table. The first set contains the market implied
expected nominal returns and covariance matrix for the Europe Value, Asia Value and
U.S. Value segments. The second set contains the market implied expected nominal
returns and covariance matrix for the Europe Growth, Asia Growth and U.S. Growth
segments. The covariance matrices are constructed by selecting the required values from
the full covariance matrix estimated in Problem 2.
Investment Theory and Applications Dr. Frederick Novomestky
FRE 6711 – Fall 2013 Final Examination
5
Problem 4 (20 points)
Use the results from Problem 3 to design the following portfolios.
 Determine the minimum variance portfolio of the Europe Value, Asia Value and U.S.
Value segments with no restrictions on short selling.
 Determine the minimum variance portfolio of the Europe Value, Asia Value and U.S.
Value segments with restrictions on short selling?
 Determine the minimum variance portfolio with target return of the Europe Value,
Asia Value and U.S. Value segments with no restrictions on short selling?
 Determine the minimum variance portfolio with target return of the Europe Value,
Asia Value and U.S. Value segments with restrictions on short selling?
Use the range of segment returns to define the target returns. That is, let min
REurope . min
RAsia
and . .
min
RU S be the minimum realized returns for Europe Value, Asia Value and U.S. Value,
respectively. Let max
REurope , max
RAsia and . .
max
RU S be the maximum realized returns for Europe
Value, Asia Value and U.S. Value, respectively. Then,  . .
min min min min R  minREurope , RAsia , RU S
and  . .
max max max max R  maxREurope , RAsia , RU S define the range of possible values for target return.
Each of the above portfolios requires that the sum of the equity segment weights is one.
What is the expected return and volatility for each these portfolios
At the 5% confidence level, compute the Value at Risk of each portfolio using the
equation 1.645 ˆ P P r   where P r is the expected return of the portfolio and ˆP  is the
estimated volatility of portfolio return where ˆ 2P
P P  w w .
Assume that the optimal weights for each of portfolios are used at the beginning of each
of the months in the six month period of January to June 2009. Using the historical
simulation method, show the realized portfolio returns for each of the portfolios.
Investment Theory and Applications Dr. Frederick Novomestky
FRE 6711 – Fall 2013 Final Examination
6

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