Posted: February 5th, 2015

Portfolios, Investments and Derivatives

Paper, Order, or Assignment Requirements

 

 

AM6200, Portfolios, Investments and Derivatives

Coursework 01: Optimal Portfolios

Instructions and Information

Assessment Type: Formal Written Coursework

Number of Questions: Answer all three questions

Module Learning Outcomes addressed: …to

• analyse the return on an asset or portfolio – as well as the mean and variance of

return – and compare investment opportunities using a variety of measures of risk,

• to identify a portfolio of assets which is optimal given a set of selection criteria,

Date Set: 10 November 2014

Date Submitted: 01 December 2014, by 10am

Initial Feedback: 08 December 2014

Final Feedback Released: 05 January 2015

Submission Instructions Submit a single hard copy of your work in the appropriate handin

box on the ground floor of the Sopwith Building (Box open 25th November 2013),

plus any electronic files in the space in the COURSEWORK section of the AM6200,

Portfolios, Investments and Derivatives module on STUDYSPACE.

Plagiarism and Academic Misconduct Ensure that all work and files that you submit are

your own. Any other such submission is classed as plagiarism.

Ensure that any work of others that appears in your submission is appropriately referenced.

If plagiarism is suspected, this counts as serious academic misconduct and will incur the

full penalty of the university.

The mark-scheme has provision available for the appropriate use of external sources.

1Questions

1. An investor has a an initial wealth w to invest and a exponential utility function: u(w) =

1 − e

−bw, where b > 0. He is given a proposal whereby for a proportion x of wealth

invested, the investment returns wealth

W =

 

w(1 + αx) with probability p

w(1 − x) with probability 1 − p

(a) Calculate the proportion x that the investor will gain maximum expected utility from

his investment.

(10 Marks)

What are the conditions on p to make the investment viable?

(3 Marks)

(b) For b = 0.001 and w = 1000, and

1. α = 1.25,
2. α = 0.75

and p = 0.6, calculate the certainty equivalents of these investment. Comment on

your answers.

(7 Marks)

(Question Total 20 Marks)

2. An investor uses the shortfall probability on the return of her portfolio as her measure of

downside risk. Her portfolio consists of a widely diverse selection of assets and its return

is assumed to have an approximately normal distribution with mean µ and variance σ

2

.

The portfolio is compared with the return on an industry index, which over the life of the

portfolio is expected to grow by 1%.

(a) Calculate the shortfall probability on the portfolio if µ = .02 and σ

2 = .005,

(3 Marks)

(b) Calculate the shortfall probability on the portfolio if µ = .018 and σ

2 = .006,

(3 Marks)

(c) Which of the above is the preferred portfolio and why?

(4 Marks)

(Question Total 10 Marks)

3. [It is assumed that you will be using an appropriate software package in this question

– which is to be submitted in the appropriate folder on STUDYSPACE. Note

also that it is your answer script that will be marked and the file will be used as

corroboration.]

A portfolio is to be constructed using the following four assets A, B, C and D.

asset A has mean return 0.02 and standard deviation .02,

asset B has mean return 0.03 and standard deviation .01,

asset C has mean return 0.04 and standard deviation .03,

asset D has mean return 0.05 and standard deviation .04.

The correlation between Assets A and B is 0.5 and between B and C is −0.2. All

other pairs of assets are uncorrelated.

(a) Calculate the vector of proportions x for the minimum variance portfolio. In your

answer, identify the theory and formulae that you are using.

(8 Marks)

(b) What are the minimum variance and the mean return of this portfolio? In your

answer, identify the theory and formulae that you are using.

(4 Marks)

(c) Calculate the portfolio giving an extra 10% on the return than minimum return (or

0 if this value is negative). What is the variance here? In your answer, identify the

theory and formulae that you are using.

(8 Marks)

(Question Total 20 Marks)

(Overall Total 50 Marks)

END OF QUESTIONS