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Due to the reputation of UTS for producing *work-ready *graduates, you are head-hunted by a small asset management firm to work part-time as a portfolio manager whilst you complete your degree. After learning all about mean-variance analysis and efficient asset allocation in 25503 Investment Analysis you are hoping to employ some of the tools you have learned in constructing your very first portfolio.

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 15% expected return target on the portfolio you construct. To get started you decide to collect historical performance data 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 ac- cording 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 15% 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:

- (a) Transform the index values into simple weekly returns (you do not need to report these in your submission).

(b) Using the returns data, calculate (and report) the vector of mean/expected returns for the five asset classes, as well as the variance-covariance matrix of these returns. These means and variances 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) re- turns ranging between 0% and 20%. 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 15% 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. You have heard many stories about commodities being great diversifiers and so you offer to investigate the effects of adding various commodities to the asset mix. Using the additional index data in the Excel spreadsheet (oil, gold, and soyabean), perform the following tasks:

- (a) Using the same methodology as in Question 1, determine which one commodity, when added to the five asset classes (giving a total of six assets), would result in the largest risk reduction of the 15% returning efficient portfolio.

(b) Report the vector of (annual) expected returns and the variance-covariance matrix for the five asset classes *plus *your chosen commodity (i.e., six assets in total).

(c) Compute and report the new *A*, *B*, *C *and ∆ parameters for this six asset portfolio. (d) Construct and plot the new MVS (with short sales allowed) for expected (annual)

returns ranging between *−*10% and 30%. You should also plot the MVS from

1.(e) for comparison and indicate the positions of the five asset classes and your chosen commodity.

(e) Determine and report the new portfolio weights for the efficient portfolio with

15% expected return.

(f ) Report the reduction in risk of the 15% returning efficient portfolio that can be achieved by adding your chosen commodity to the portfolio. (Remember this should be larger than for the other two commodities that were not chosen.)

You inform your boss of these findings but she decides not to include the extra commodity into the portfolio as she is concerned about the future performance of commodities. She has also realised that the 15% returning portfolio you have constructed is not as ‘efficient’ as it might be because you have forgotten all about the risk-free asset… oops! Rather than a commodity, the risk-free asset should be added to the original portfolio of five asset classes. 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 from Question 1:

- (a) Construct and plot the MVS (with short sales allowed) for the five asset classes
*plus*a risk-free asset paying 1%. (Note: do not included your chosen commodity from Question 2.)

(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 1.(e) on the same set of axes.

(c) Determine and report the new portfolio weights for the efficient portfolio with

15% expected return.

(d) Calculate and report the reduction in risk of the 15% returning efficient portfolio that can be achieved by adding the risk-free asset to the portfolio of five asset classes.

A little embarrassed from your mistake of not including the risk-free asset in the first place, 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 portfolio. 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:

- (a) Construct and plot the risky asset only MVS with no short sales allowed for the five asset classes (i.e., without the chosen commodity from Question 2 or the risk-free asset from Question 3). Recall that you will need
*Solver*to do this.

(b) Plot the MVS for the unconstrained problem—found in 1.(e)—on the same set of axes. Also, indicate the positions of the five asset classes on your figure.

(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 con- straint 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.

To be continued…