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It is your ﬁrst day on the job and your boss is keen to see how much you really know. She provides you with a list of ﬁve asset classes and tasks you and your team to investigate the eﬃcient asset allocation between these asset classes. Moreover, you are asked to satisfy a 17% expected return target on the portfolio you construct. To get started you decide to collect historical performance data for the last ﬁve years in order to estimate the expected return and variance-covariance structure of the asset classes (the data in the Excel ﬁle).
To perform the asset allocation you decide to construct a minimum variance portfolio according to the theory you learned in 25503 Investment Analysis. After all, what can go wrong? Harry Markowitz won a Nobel prize for this stuﬀ! You recall the 17% expected return target imposed by your boss and note that there was no mention of short-selling constraints. In order to construct this portfolio you should copy the assignment data into an Excel workbook and perform the following tasks/answer the following questions:
1. (a) Transform the index values into simple weekly returns (you do not need to report these in your submission). (b) Using the returns data, estimate (and report) the vector of expected returns for the ﬁve asset classes, as well as the variance-covariance matrix of these returns. These expected returns etc. should be annualized (i.e., in annual units). (c) Report which of the asset classes are eﬃcient and which are ineﬃcient. For each of the ineﬃcient asset classes, ﬁnd another asset class that dominates it. (d) Compute and report the parameters A, B, C and ∆. (e) Construct and plot the MVS (with short sales allowed) for expected (annual) returns ranging between −10% and 35%. Your ﬁgure should also indicate the positions of the ﬁve asset classes. (f) Identify the global minimum variance portfolio (MVP), i.e. report the portfolio weights (in the ﬁve asset classes), expected return, and variance of the MVP. (g) Determine and report the portfolio weights for the eﬃcient portfolio with 17% expected return.
You are eager to impress so you send the results to your new boss just before you leave for your lunch break. Upon your return, the boss has looked at your report and notes that the risk of the portfolio is a little higher than she expected and wondered if adding an additional asset would help reduce the risk. Knowing all about diversiﬁcation, you suggest that maybe adding an asset that has a low correlation with the existing ﬁve asset classes might help. “Gold!”, your boss exclaims. You have heard many stories about gold being a great diversiﬁer and so you oﬀer to do the analysis again with the additional asset (gold) included in the portfolio. Using the additional gold index data in the Excel spreadsheet, perform the following tasks:
2. (a) Using the same methodology as in Question 1 (simple returns etc.), reconstruct the vector of (annual) expected returns and the variance-covariance matrix for the ﬁve asset classes plus gold (i.e., six assets in total). (b) Compute and report the new A, B, C and ∆ parameters. (c) Construct and plot the new MVS (with short sales allowed) for expected (annual) returns ranging between −10% and 35%. You should also plot the MVS from 1.(e) for comparison and indicate the positions of the ﬁve asset classes and gold. (d) Identify the new global minimum variance portfolio (MVP), i.e. report the portfolio weights (in the six assets), expected return, and variance of the MVP. (e) Determine and report the new portfolio weights for the eﬃcient portfolio with 17% expected return. (f) Calculate and report the reduction in risk of the 17% returning eﬃcient portfolio that can be achieved by adding gold to the portfolio.
You inform your boss of these ﬁndings and she is happy with the addition of gold to the portfolio and the reduction in risk. However, she informs you that the 15% returning portfolio you have constructed is not as ‘eﬃcient’ as it might be as you have forgotten all about the risk-free asset… oops! You quickly do some research and determine that the appropriate risk-free rate to use is 1% per annum. Perform the following tasks to adjust your portfolio weights:
3. (a) Construct and plot the MVS (with short sales allowed) for the ﬁve asset classes and gold plus the risk-free asset paying 1%. (b) Identify the tangency portfolio, i.e. report its portfolio weights, expected return, and variance of returns. Furthermore, illustrate its tangency property graphically by plotting the MVS from 2.(c) on the same set of axes. (c) Determine and report the new portfolio weights for the eﬃcient portfolio with 17% expected return. (d) Calculate and report the reduction in risk of the 17% returning eﬃcient portfolio that can be achieved by adding the risk-free asset bond to the portfolio of six risky assets.
A little embarrassed from your mistake of not including the risk-free asset, you send the new updated results to your boss at 4:50pm. She is impressed with your eﬃciency as well as the eﬃciency of the portfolio. However, she hasn’t quite ﬁnished with you just yet! She is worried about the need to short sell certain asset classes in the currently proposed portfolios. Many of the ﬁrm’s clients do not like, and some do not allow, short selling in their portfolios. Therefore, your boss wants you to investigate the eﬀect a no short sales constraint will have on the MVS without a risk-free asset and any subsequent investment decisions. To do this you are asked to perform the following tasks:
4. (a) Construct and plot the risky asset only MVS with no short sales allowed for the ﬁve asset classes plus gold. (Recall you will need Solver to do this.) (b) Plot the MVS for the unconstrained problem—found in 2.(c)—on the same set of axes. Also, indicate the positions of the ﬁve asset classes plus gold. (c) List the portfolio weights for all the data points used in constructing your no short sales allowed graph. (d) Identify and report the range of expected returns for which the short sales constraint is not binding. (e) Discuss the compositions of the portfolios at the end-points of the MVS with no short sales.
At 7:15pm, with a grumbling stomach, you send the results to your boss who is still working hard in her oﬃce. As you gather your things to leave, your email pings and it is a lengthly reply from your boss, outlining yet further questions and instructions… however, your boss has kindly said that your reply can wait until tomorrow.
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