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1 ECMT2130 - 2021 semester 1 mid-semester exam 1 solutions

Author: Geoﬀ Shuetrim

1. (?? points) Portfolio optimisation data

Andrea is able to invest in 4 risky assets. Using a sample of data she has estimated the average simple monthly rates of return and the variances/covariances of those simple monthly rates of return for the various risky assets.

Her estimates are contained in columns A to N of this Excel spreadsheet (Spreadsheet provided sepa- rately), along with the correlation matrix implied by the variance and covariance estimates. She has also included CAPM Beta estimates for each of the assets.

Use the information in the spreadsheet to answer the following related questions in your exam. Use the spreadsheet to perform the necessary calculations. Document the calculations clearly in the spread- sheet by placing informative labels next to cells that contain important formulae to ensure that your calculations are easy to review.

Upload your ﬁnal Excel spreadsheet, with all of the original data, and your calculations for related exam questions, as part of your exam response.

Make sure that your solver conﬁguration is clear in the uploaded ﬁnal Excel spreadsheet for for each part of the question 2. (use a diﬀerent worksheet in the Excel workbook for each solver conﬁguration you use - but you should only need 1)

The spreadsheet formulae and solver conﬁguration will be reviewed as part of assessing your marks for the part(s) of question 2 involving optimisation calculations.

Solution:

Question 1 just supplies data for question 2.

2. (?? points) Andrea’s portfolio optimisation analysis

As your response to this question, your handwritten or typed answer to all parts needs to be uploaded as a single ﬁle (a photo / a MS Word document / or a PDF).

Answer this question using the data from question 1. Do not spend time using algebra to solve these problems. Instead solve it numerically in MS Excel.

You should only need to use the Excel solver for part A.

(a) (5 points) As a percentage, what is the expected rate of return on the fully invested portfolio of risky assets that maximises its expected rate of return while ensuring that its return standard deviation

is no more than the minimum return standard deviation among the individual risky assets? (b) (1 point) Is the portfolio from part A eﬃcient? Explain your answer.

(c) (2 points) Write out in full, the deﬁnition of the problem that you have solved to determine the weights on the portfolio with least risk. Include, the objective function, the variables that are being adjusted to minimise that objective function, and the constraints that limit how those variables can be adjusted.

(d) (3 points) Explain the weights on the risky assets in the portfolio that solves the problem deﬁned in part A. In your explanation, draw upon the information about risky asset returns in the spreadsheet from question 1.

(e) (2 points) Could the investor hold an eﬃcient portfolio of the risky assets that had an expected return no less than 20%? Explain your answer.

(f) (1 point) Could the investor hold a portfolio of the risky assets that had an expected return no less than 20% without shorting any of the risky assets? Explain your answer.

(g) (1 point) In practice, what is the qualitative diﬀerence between the risk associated with holding a long position in an asset, compared to the risk associated with holding a short position in an asset?

Solution:

(a) In the solution portfolio, the expected return is 1.88%. This is achieved by setting the weights on assets 1, 2 and 3, and 4 to 0.09, -0.07, and 0.64, and 0.34 respectively, rounded to 2 decimal places.

(b) The portfolio is eﬃcient. It has minimum variance for a given required expected return and the variance is greater than the GMVP portfolio variance. This places it on the eﬃcient frontier given the 4 available assets.

(c) We are maximising the expected return of the portfolio, w′ E(r) or E(r)′ w, by changing the weights on the risky assets while ensuring that two constraints are satisﬁed: full investment, requiring the weights to sum to 1, and portfolio return standard deviation is no greater than the minimum return standard deviation across all risky assets.

(d) We have weights that correspond to a point on the eﬃcient frontier so the weights should adapt to ensure positive weights on assets with higher expected returns per unit of standard deviation. Negative weights on assets that are positively correlated with assets that we are long on (to maximise expected return). Positive weights on assets that are negatively correlated with assets that we are long on (to maximise expected return). More speciﬁcally, the highest weights are on assets 3 and 4, which have the highest expected returns (which we are maximising). The weight on asset 2 is negative, exploiting its positive correlation with assets