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**25503 Investment Analysis**

In this assignment you will implement an investment strategy (along with other tasks) based

on mean-variance portfolio theory. In order to help you do this you will nd an Excel

workbook called AssignmentData.xlsx on UTS Online. It contains weekly index data for

ve asset classes and one individual asset (Gold).

1

Due to the reputation of UTS for producing work-ready graduates, you are head-hunted by

a small asset management company to work part-time as a portfolio manager whilst you

complete your degree. After learning all about mean-variance analysis and ecient asset

allocation in 25503 Investment Analysis you are hoping to employ some of the tools you

have learned in constructing your very rst portfolio.

It is your rst day on the job and your boss is keen to see how much you really know.

She provides you with a list of six asset classes and tasks you to investigate the ecient as-

set allocation between these asset classes. Moreover, you are asked to maximise the expected

utility of a client, which is measured by p 1

2a2

p, where the level of risk aversion a = 5.

To get started you decide to collect historical performance data for the last ve years in

order to estimate the expected return and variance-covariance structure of the asset classes

(the data in the Excel le).

To perform the asset allocation you decide to use the mean-variance portfolio theory you

learnt in in 25503 Investment Analysis. After all, what can go wrong? Harry Markowitz won

a Nobel prize for this stu! However, it has been a whole year since you completed the sub-

ject, so naturally you feel a bit rusty. You recall the lecturer mentioning about ecient and

inecient assets, and that rational investors should never choose to invest in any inecient

assets. So, rst you decide to nd out which of the six assets are ecient and which are

inecient, then construct a portfolio of the ecient assets. In order to do that, you should

copy the assignment data into an Excel workbook and perform the following tasks:

1. (a) Transform the index values into continuously-compounded weekly returns (you do

not need to report these in your submission).

(b) Using the resulting returns data, estimate (and report) the vector of expected

returns for the six assets, 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 assets are ecient and which are inecient. For each of the

inecient assets, nd another asset that dominate it. If the client was to invest

100% of her wealth into one of the ecient assets, which one should she choose?

(d) Choose two of the ecient assets, construct and plot the combination line between

them (with short sales allowed) for expected (annual) returns ranging between 0%

and 20%. Your gure should also indicate the positions of the six assets.

(e) Identify the minimum variance portfolio (MVP), i.e. report the portfolio weights

(in the two ecient assets), expected return, and standard deviation of the MVP.

(f) Based on the client’s utility function, what is her optimal portfolio? i.e. report the

portfolio weights, expected return and standard deviation of her optimal portfolio.

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

2

risk of the portfolio is a little higher than she expected and wondered if adding additional

assets would help reduce the risk. Of course! You suddenly remember that the inecient

assets can still be useful if we include them in a portfolio, that is what mean-variance portfolio

theory and diversication is all about! You quickly run back to your desk and perform the

following tasks:

2. (a) Using the vector of expected returns and variance-covariance matrix computed in

Question 1(a), compute and report the A, B, C and parameters.

(b) Identify the global minimum variance portfolio (GMVP), i.e. report the portfolio

weights (in the six assets), expected return, and standard deviation of the GMVP.

Compared to the MVP in Question 1(e), are there any improvements?

(c) Determine and report the portfolio weights and standard deviations for two min-

imum variance portfolios (MVPs), P1 and P2, which have the same expected

returns as the two ecient assets in Question 1(d). What are reductions in stan-

dard deviations?

(d) Construct and plot the combination line between P1 and P2 (with short sales

allowed) for expected (annual) returns ranging between 0% and 20% (note that

this is also the MVS with short sales allowed). You should also plot the combina-

tion line between the two ecient assets from Question 1(d) for comparison, also

indicate the positions of the six assets and the two MVPs.

(e) Now based on the client’s utility function, what is her optimal portfolio (i.e.

portfolio weights in the two MVPs, expected return and standard deviation)?

Compare to the answer in Question 1(f), is she now better o? Report the

increase in her expected utility.

You inform your boss of these ndings and she is happy with the reduction in risks and

the improvement in the client’s expected utility. However, she informs you that the 10%

returning portfolio you have constructed is not as `ecient’ 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 3% per annum. Perform the following tasks to

adjust your portfolio weights:

3. (a) Using the A, B, C and parameters in Question 2(a), compute the expected

return and standard deviation of the tang ency portfolio.

(b) Using the Two Fund Theorem, determine the proportions that need to invested

in P1 and P2 to form the tangency portfolio.

(c) Construct and plot the MVS (with short sales allowed) for the six assets the risk-

free bond paying 3%. Furthermore, plot the combination line between P1 and P2

from Question 2(c) on the same set of axes, also indicate the positions of the six

assets, two MVPs and the tangency portfolio.

3

(d) Determine and report the new optimal portfolio for the client, i.e. the portfolio

weights in the risk-free bond and the tang ency portfolio, expected return and stan-

dard deviation. Compare to Question 2(e), what is the increase in her expected

utility?

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 eciency as well as the

eciency of the portfolio. However, she informs you that due to company’s credit worthiness,

any leverage (i.e. borrowing at the risk-free rate) and short-selling are not allowed. But you

just did all that work…oh well. Your boss wants you to investigate the eect 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

six assets. (Recall you will need Solver to do this.)

(b) Plot the combination line between P1 and P2 in Question 2(c)|on the same set

of axes. Also, indicate the positions of the six assets.

(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.

(f) Now with no short sales allowed, what is the optimal portfolio for the client?

Report the portfolio weights in the six assets, expected return and standard de-

viation of her optimal portfolio. Also, compare to Question 2(f), what is the

reduction in client’s expected utility?

At 7:15pm, with a grumbling stomach, you send the results to your boss who is still working

hard in her oce. As you gather your things to leave, your email pings and it is a lengthly

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