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1 ECMT2130 - 2021 semester 2 mid-semester exam 1 solutions

Author: Geoﬀ Shuetrim

1. (0 points) Portfolio optimisation data

David is able to invest in 3 risky assets. Using a sample of data he 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.

His estimates are contained in columns A to H of this Excel spreadsheet.

Use the information in the spreadsheet to answer the various parts of question 2 below. Use the spread- sheet to perform the necessary calculations. Document the calculations clearly in the spreadsheet 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. Make sure that you use a diﬀerent worksheet in the Excel workbook for each

solver conﬁguration you use - but note that you should only need to use the solver once. The spreadsheet formulae and solver conﬁguration will be reviewed as part of assessing your marks for the parts of question 2 involving optimisation calculations.

Solution:

Question 1 just supplies data for question 2.

2. Portfolio optimisation question

As your response to this question, your handwritten or typed answer to all parts needs to be uploaded. Answer this question using the data from question 1. Do not spend time using algebra to solve these problems. Instead solve the various problems numerically using Microsoft Excel.

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

(a) (4 points) What are the weights on risky assets in the portfolio that is on the optimal capital

allocation line while having a weight of 0 on the risk-free asset (the tangency portfolio)? (b) (2 points) What is the expected return for the tangency portfolio?

(d) (3 points) What is the value of the Sharpe Ratio for the tangency portfolio?

An investor who maximises expected utility has expected utility function E(U) = E(rp ) − 0.5Aσp(2) where rp is the return on the portfolio held by the investor and sigmap is the return standard deviation for that portfolio. A is a positive constant that determines the investor’s risk aversion.

(e) (2 points) In terms of A and σp , what is the slope of the investor’s indiﬀerence curve, obtained from the expected utility function by treating E(U) as a constant and rearranging so that E(rp ) is a function of σp ?

(f) (2 points) What is the value of A that would lead this investor to invest in the tangency portfolio,

without investing in the risky asset at all?

Solution:

(a) In the tangency portfolio, the weights on assets 1, 2 and 3 are -0.55, -0.26, and 0.36 respec- tively, rounded to 2 decimal places. To get full marks the spreadsheet solver had to be set up appropriately and run to obtain the weights that were reported. In particular, the objective needed to be right, the weights being optimised needed to be correct (excluding the weight on the risk-free asset), and the full investment constraint had to be imposed. Short selling also had to be allowed.

(b) The expected return for the tangency portfolio is 3.39%. To get full marks, it was necessary to include the formula computing this value in the uploaded Excel spreadsheet.

(c) The return standard deviation for the tangency portfolio is 12.42%. To get full marks, it was necessary to include the formula computing this value in the uploaded Excel spreadsheet.

(d) The Sharpe Ratio for the tangency portfolio is 24.89%. To get full marks, it was necessary to include the formula computing this value in the uploaded Excel spreadsheet.