Uploaded by Max Rose

Problem Set #3

Problem Set #3
This problem set is due at the beginning of class (2:00pm ET). You may collaborate with your
classmates in preparing the solutions, but each student must hand in a separate assignment.
Assignments should be submitted through Brightspace. Assignments that are late but within 24
hours of the deadline, will receive ½ credit. After 24 hours no assignments will be accepted
(unless due to documented serious illness or family emergency)!
Download the Excel spreadsheet “Problem Set #3” from Brightspace. Please put your name and
net ID in the blue boxes at the top of the worksheet. This spreadsheet contains monthly (US
dollar) returns for the period Jan. 1988-Dec. 2020 on the US, MSCI EAFE (Europe, Australasia,
and the Far East developed markets), MSCI EM (emerging markets) and MSCI ACW (All
Country World) equity indices, and the corresponding monthly risk-free rate. (Note that the first
3 indices are exactly the same series as you used in Problem Set #2 so you have already done
some of the work.) The cells where formulas need to be added are highlighted in yellow.
Because your solutions will be computer-graded it is very important to follow the instructions:
• Please do NOT reformat the file or move the yellow cells. Do NOT add rows or columns
to the worksheet. However, you can add additional calculations in non-highlighted cells if
you wish. Everything in non-highlighted cells will be ignored for grading purposes. You
can either enter the formula to calculate your answer in the yellow cells, or enter these
formulas elsewhere and just put references to these answers in the yellow cells.
• Please do NOT round your answers or intermediate calculations. Excel will display
rounded answers based on the formatting of the cells, but it will retain the unrounded
• Please do NOT add a new worksheet to the file. Do all of your calculations in the “Results”
worksheet and submit a file with only a single worksheet.
1. Follow the steps below to generate inputs for your portfolio optimization:
a. Calculate monthly excess returns (in excess of the risk-free rate) for each index. Put these
numbers in the four marked columns.
b. Calculate the averages, standard deviations and correlations of the monthly excess returns
on the 4 indices. Annualize the averages and standard deviations, calculate the annualized
Sharpe ratios and the annualized average total returns (using the annualized risk-free rate
of 0.5% given in cell T20).
c. Calculate the alphas and betas of the US, EAFE, EM, and ACW indices relative to the
ACW index by running regressions using the monthly excess returns. Instead of using the
regression procedure built into the Analysis ToolPak, just use the =SLOPE and
=INTERCEPT functions because we will not be worrying about standard errors, tstatistics, or R-squareds for this exercise. Also, calculate the annualized alphas (multiply
the monthly alpha by 12). (Note: The betas do not need to be annualized; they should be
the same regardless of the frequency of the data.)
d. To get the expected total return inputs for the 4 indices, assume the CAPM holds with the
ACW as the market portfolio. Use the estimated beta from part (c), an annualized market
risk premium equal to the average annualized excess return on the ACW index, and an
annualized risk-free rate of 0.5%. (Run your calculations directly off the unrounded
market risk premium in cell T27, which is just the annualized average excess return in
R17. Do NOT put a rounded number in these calculations.)
To think about: How are these model-based expected returns related to the historical
average returns and the historical alphas?
2. Consider the US, EAFE, and EM.
a. Calculate the (unconstrained) portfolio weights for the minimum variance and maximum
Sharpe ratio portfolios using the expected returns from part 1(d) and the historical
volatilities and correlations.
To think about: What does the minimum variance portfolio look like relative to the
unconstrained minimum variance portfolio from PS#2? What is the Sharpe ratio of the
maximum Sharpe ratio portfolio relative to the Sharpe ratio of ACW? Under the CAPM
what is the maximum Sharpe ratio portfolio?
b. Calculate the covariance between the minimum variance and maximum Sharpe ratio
portfolios from part a above (just as in PS#2). Use either the long (9-term) formula or the
short (linear algebra) version. (See the Session 4 lecture notes.) Note that the covariance
matrix (Σ) will automatically be calculated for you from the numbers you compute in Q.1
and the matrix for the US, EAFE and EM appears in the range V17:X19. The full 4x4
matrix in V17:Y20 includes ACW and will be used below.
c. Calculate the correlation between these 2 portfolios.
d. Calculate expected returns and standard deviations of the portfolios on the 3-risky asset
frontier by computing these numbers for combinations of the minimum variance and
maximum SR portfolios. (The weight in the minimum variance portfolio varies from
-230% to 390%.) If you do this correctly the IOS will appear in the graph. Also compute
the Sharpe ratio for each portfolio.
3. Finally consider all 4 series: the US, EAFE, EM, and ACW. Compute the maximum Sharpe
ratio portfolio using all 4 assets, again using the CAPM implied expected returns from 1(d).
Note that the variance of a 4-asset portfolio is a somewhat long and messy formula.
However, if you are willing to use the linear algebra version, it is much shorter. For this latter
calculation you need the full covariance matrix in the range V17:Y20.
To think about: What do you notice about the weights in this portfolio? Does this make
sense? Why?