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Homework Assignment 3

ECN382 Portfolio Management

In this homework, you will implement the Black-Litterman model.

Homework Data

Data for this assignment is provided in two .txt files, “5_Industry_Portfolios” and “F- F_Research_Data_Factors”, both of which can be downloaded from QMplus.

These datasets are from Kenneth French’s website, which contains a variety of useful datasets on asset markets. For your reference, the two data files used for this assign- ment can be located on the website as follows:

• 5 Industry Portfolios: there is a section entitled “Industry Portfolios” about halfway down the page, and the link to the file used is in the first row in that section.

• Fama/French 3 Factors: the link to this is in the first row of the section entitled “U.S. Research Returns Data (Downloadable Files)”, which is just below the table on the top of the page.

Please Note: the download instructions for the website are provided for your ref- erence only (the dataset is widely used in Finance, so it’s useful to know where the data comes from). For purposes of the homework, however, please use the files pro- vided on QMplus, and not your own download. Sometimes, the data provided by Ken French changes (even retroactively!), and if you download the data more recently than I did it is possible that it might have changed.

We will work with monthly returns data, which is found in the first section of each  of the two files. From the first one, we would like returns data for the five sectors in the US equity markets, which are Consumer, Manufacturing, Technology, Healthcare and Other. From the second, we will only need the risk-free returns, which is the last column of the dataset (the series named “RF”).

For this homework, we will use returns data from January 1980 to December 2019.

We will also need data for market capitalisation for each sector. While this data is not provided directly in the downloaded file “5_Industry_Portfolios”, it can be computed from the data in it (use the “‘Number of Firms in Portfolios” and “Average Firm Size”). In order to compute market capitalisation in the homework, please use data as of the end of the sample, that is, for December 2019.

Please also note that the returns data provided in both files is in percentage terms. For example, the Consumer Sector returns for the month of July 1926 is given as “5.43”. This means that the return was 5.43%. Thus, in order to work with the data from these files, you will need to divide every return number by 100.

Using this data, complete the following five tasks:

1. Construct the annualised covariance matrix of the sector excess returns for the entire sample. Also compute market capitalisation sector weights as of Decem- ber 2019. With this data, compute the implicit vector Π of expected excess re- turns for the sectors according to our implementation of Black-Litterman (as- sume that λ = 4).

2. Compute the equilibrium weights for the investor, assuming that the prior as- sumption on the mean returns has been imposed, but no views have been in- corporated into the portfolio (assume that τ = 0.05). (Recall that we call these weights the prior weights).

Next, we will use the Black-Litterman framework to compute the optimal portfolio weights given the following two views:

• The Consumer sector is expected to have an excess return of 9% over the next year;

• The Technology sector is expected to outperform the Healthcare sector by 6% over the next year

We also need information on the variance of these views. To determine this variance, please use the following information:

• The absolute view on the Consumer sector has an annualised standard deviation of 10%

• The relative view on the High Tech sector versus the Healthcare sector by 6% has an annualised standard deviation of 12%

• The two views are uncorrelated

3. Express the matrices P, Q and for these two views. With this information, compute the equilibrium weights that include your views. Also compute the equilibrium weights relative to the prior weights computed in Item 2 (i.e., com- pute the difference between the equilibrium weights in Item 3 and the prior weights in Item 2.)

We now change the confidence level in our views by increasing the standard devia- tions associated with them. Namely, we maintain the assumptions regarding expected excess returns in the views, but replace those associated with the standard deviations of the views by the following new assumptions:

• The absolute view on the Consumer sector has an annualised standard deviation of 15%

• The relative view on the High Tech sector versus the Healthcare sector by 6% has an annualised standard deviation of 20%

• The two views are uncorrelated.

4. Express the updated matrix for the views. With this information, compute the equilibrium weights that include your views. Also compute the equilibrium weights relative to the prior weights computed in Item 2.

5. Compare the weights you found in items 2, 3 and 4. Do they make sense in light of what we learned about the properties of the Black-Litterman model?

The last item requires you to submit a written answer to a problem. The answer can be handwritten clearly or typed. Please convert it to .pdf format before submitting it.

6. This item requires you to follow the notation of the Black-Litterman model in the lecture notes. Follow the steps of the Black Litterman model discussed therein and show that the Black-Litterman algorithm will yield an optimal portfolio

given by w = wCAP /(1 + τ ) (where wCAP is the vector of market-capitalisation weights) if:

(i) you hold no private views, that is, the entries in P and Q matrices are zeros; and

(ii) you believe that the market consensus is on average correct about µER, but has a measurement error. Specifically, µER (Π, τ Σ), so that µER is nor- mally distributed with a mean vector Π and a variance matrix of τ Σ.

Submitting Your Work:

1. The “Homework Assignment 3” Excel file contains a sheet named “Answers”. Please provide your answers to each question above in the appropriate section of this sheet. Do not modify this sheet in any other way.

2. Please submit your supporting work in additional sheets of your submission file. Include as much information as needed for me to verify the correctness of your work. Submissions that do not include appropriate supporting work incur a substantial mark deduction.

3. In your answer sheet, please enter your Exam Candidate ID Number in the indi- cated range. Do not use your name, including in the filename.

4. For item 6, as instructed above, please write up your answers and submit them as .pdf file along with your submission Excel file.

Your work must be submitted through QMplus (not via email) The deadline is Wednes- day, November 23, at 11:59pm. No extensions will be granted.