Portfolio Management

Construct your dataset: Your dataset will consist of the monthly arithmetic returns R_i,t of 3 stocks i={1,2,3} listed in an exchange of your choice and the monthly returns R_m,t of a financial market index of the exchange you have considered. For example, if you choose stocks traded in LSE, the FTSE 100 index can be used as a market index. You can download these data from the BLOOMBERG database (room Rf39) or from a free data provider like YAHOO!FINANCE. The requested period is from January 2012 until December 2015 (4 years). In particular, you must split your dataset into 2 sub-periods of 2 years each (see Table 1).

Table 1: Breaking your dataset into two sub-periods

Answer Questions 1-4 for the 1^st sub-period.

1. Calculate the expected (mean) return (R_i) the variance (σ_i²) and the standard deviation (σ_i) for each security. In addition calculate the correlation matrix and the variance/covariance matrix. Use the historical data for your calculations.

2. Create 20 randomly generated portfolios (j={1,2,...,19,20}) by using random weights to each of the securities you considered (X_i). Assume that short sales are not allowed. Then calculate the expected return (R ¯_(P,j)) and the standard deviation (σ_(P,j)) of these portfolios. On a 2D system of coordinates (y-axis: expected returns, x-axis: standard deviation) illustrate those portfolios along with the initial stocks you have considered.

3.Assume that short sales, riskless lending and borrowing are allowed. Also assume that the lending rate equals to the borrowing rate (R_F).
  a. Set a value for R_F and justify your choice.
  b. Derive the optimal portfolio (composition, expected return and standard deviation)
  c. Derive the efficient frontier.

4. Assume that the Sharpe single-index model describes the correlation structure of security returns. Using the single-index model, calculate the following quantities:

The alpha and the beta for each stock.

The standard deviation of the residuals from each regression.

The expected return and the variance for each security.

The covariance between each possible pair of stocks.

The expected return and the standard deviation of a portfolio by using the weights of the optimal portfolio you found in 3.b.

Compare the results you found in sub-questions 4.c, 4.d and 4.e with those found in question 1 and the sub-question 3.b. Comment any differences or equalities that you found on the above figures.

What is your best forecast of beta for each stock, for the 2nd sub-period by using historical data of the 1st sub-period?

In the next question you need to consider the 1st and the 2nd sub-periods.

5. Calculate the beta for each stock for the 2nd sub-period and compare with the forecast you made in question 4.g. What techniques can be adopted to adjust your beta predictions which are based on historical estimates? Explain briefly.

Further important notes:
1. This assignment contributes 30% to your final mark.

2. Please comment on the results you find from your calculations.

3. Please ensure that any source (i.e., article, book, etc) that you have cited (if any) in the text appears in your reference list. The 4. Harvard referencing style should be applied.

4. If you need any further information or help please contact me via email or come by my office (Rf12) during my office hours (Mondays & Tuesdays 10:00-11:00).

5. You are allowed to use any software you prefer (e.g. Ms Excel, R, etc.). The submission will be done via blackboard and you must submit at least two files.

  a. The first file which is your report (word document or pdf) should present all your results (including tables and figures). It is expected that you comment on all your results.

  b. The second file that you will submit depends on the software that you will use. For example, if you use Ms Excel, you must submit an *.xlsx file containing your dataset along with your calculations.

More information will be provided regarding the submission process. In addition, more guidelines will be given during lectures.

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