Portfolio Management
Homework 1 SOLVED
Answer the following questions by copying and pasting results in the appropriate
space. Use the same spaces to write comments when required.
Instructions: With reference to the entire homework, suppose that there are only two
risky assets: equity and corporate bonds (please take the series from the data set that was
introduced in lectures 4 and 5, in class). Do not impose the constraint that the weights
should be positive, unless this is clearly stated in the questions. Questions 1 through 4
require you to copy and paste your results in this word file. Question 5 requires you to
also hand in your excel worksheet to the instructor for grading. By the stated deadline
(15th April 2017), please send an email to Prof. Guidolin (at
[email protected]), to include:
a) Word or pdf file containing your answers;
b) Excel file containing your calculations in question 5.
Both files should contain your name and surname in the file denomination. No exceptions
– failure to indicate your name will mean to imply you want to remain anonymous and
receive zero grade.
1. Please compute the annualized means of, the correlation between the two asset
returns, and the annualized variance-covariance matrix and copy and paste the results in
the space below.
2. Please compute the annualized mean, variance and standard deviation of a portfolio
composed by 40% of equity and 60% of corporate bonds and paste your results in the space
provided below.
3. Please compute the weights of the global minimum variance portfolio (GMVP). Please
paste the mean, variance and standard deviation of the GMVP returns in the space provided
below. In addition, plot in the risk-return (standard deviation/expected return) space: a) a
portfolio composed by 100% of equity; b) a portfolio composed by 100% of corporate bonds;
c) the GMVP. Is there any chance that a rational, non-satiated investor will invest 100% of
her wealth in corporate bonds? Briefly motivate your answer.
A rational, non-satiated investor will not invest 100% of her wealth in corporate
bonds. A 100% corporate bond is on the mean-variance (MV) frontier, but is NOT on
the efficient frontier, as it is dominated by the Global Minimum Variance Portfolio
(GMPV). Recall that the efficient frontier is the portion of the MV frontier that starts
from the GMVP.
4. Please compute the weights of a portfolio with mean return equal to 10% that lies on the
efficient frontier and paste them in the space provided below. Would it be possible to achieve
a portfolio with mean return equal to 10% and positive weights (i.e., no short-selling)? Please
make sure to clearly justify your answer.
Clearly, the maximum return that can be achieved without short-selling is 9,06%, i.e.,
the return of a portfolio that is totally invested in equity. In order to achieve a higher
return, it is necessary to short-sell corporate bonds and invest the money so obtained
into equity.
5. Use a simple bootstrap approach with 1,000 alternative scenario of (simple) returns
between time t and t+1. Compute means, variances, standard deviations, and the pair-wise
linear correlation from the 1,000 simulations. Compute the optimal weights for an investor
whose preferences are characterized by a mean-variance utility function with: a) parameter
kappa equal to 0.2 and b) kappa equal to 0.5. Please hand in your excel computations in the
form of a file and provide below a short explanation of the difference between the two sets
of weights obtained under (a) and (b).
See the excel from lecture 5 – you were only required to change the data. Note: as
discussed during the lecture, 1000 simulations may not be enough to guarantee the
stability of the results, therefore you may have (correctly) obtained results that are
different from your classmates. Only the procedure was graded, with no attention to the
numbers concerning the resulting portfolio weights.
Whatever exact weights you obtained, a coefficient k=0,2 must have been associated
with a higher portion of equity than the one associated with a coefficient k=0,5 (see slide
copied below). This follows frome the fact that as k increases, risk tolerance decreases.