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

Author: Geoﬀ Shuetrim

1. (0 points) Portfolio optimisation data

John 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 G 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. There will be several versions of question 1.

2. (15 points) John’s portfolio optimisation analysis

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 and has a weight of 50% on the risk-free asset (we will refer to it as the half-way portfolio)?

(b) (2 points) What is the expected rate of return for the half-way portfolio?

(d) (3 points) What is the value of the Sharpe Ratio for the half-way portfolio?

An investor who maximises expected utility has expected utility function E(U) = E(rp ) - 0.5Aσp 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 level of risk aversion.

(e) (2 points) What is the value of A that would lead this investor to invest in the half-way portfolio?

(f) (2 points) If you allowed the investor with your computed value of A to invest in a 4th risky asset,

qualitatively, how would that change the Sharpe Ratio of the investor’s new optimal portfolio?

Solution:

(a) In the half-way portfolio, the weights on assets 1, 2 and 3 are 0.58%, 9.05%, and 40.36% respectively, rounded to 2 decimal places. These are half the weights on the three assets in the tangency portfolio.

(b) The expected return for the half-way portfolio is 1.17%

(d) The Sharpe Ratio for the half-way portfolio is 0.33.

(e) Diﬀerentiating the expected utility function, we obtain:

dE(rp )

For any optimal portfolio for the investor, this gradient will be equal to the slope of the Capital Allocation Line (the Sharpe Ratio), S . Thus,

Aσp = S = 0.33

If the optimal portfolio for the investor is the half-way portfolio, then its return standard deviation for that portfolio will be σp . Thus, we can substitute 3.22% for σp and solve for A:

A = 0.33/0.0322 = 10.25

(f) The Sharpe Ratio would likely increase and certainly not decrease with the addition of the new risky asset because it increases the range of investment options available to the investor. This will tend to shift the eﬃcient frontier (for risky assets) to the left, reﬂecting the increased diversiﬁcation options. This will cause t