Posted: January 13th, 2023

AcF 351b: Practical Portfolio Optimization project:

AcF 351b: Practical Portfolio Optimization project:
You are in charge of the equity portfolio for the investment company Bronzeman Sachs Ltd
and should develop a new investment strategy that provides a good risk-return trade-off. In
particular, you should try to find a portfolio that will outperform the market index.
The growing interest on smart beta funds has forced your firm to offer a smart beta product.
This product will be formed as the combination of a particular investment style with the
market portfolio. You can find your assigned investment style in a separate pdf named
“Students_351b_Investment_Style_22_23.pdf”. Further, you can find the returns of the
market portfolio and your assigned investment style in the excel file named
“Styles_data_22_23.xlsx”. Your objective is to combine the market with your assigned
investment style such that the combined returns deliver a better risk-return trade-off than
the market.
In your report, respond, in order, to each of the following tasks:
1. Plot the cumulative returns of the market portfolio and your assigned investment
style. Ace my homework – Write my paper – Online assignment help tutors – Discuss the results from this graphical representation. [10 marks]
2. Compute and report the Sharpe ratio of your assigned investment style using the
entire sample. Ace my homework – Write my paper – Online assignment help tutors – Discuss this result relative to the Sharpe ratio of the market portfolio.
[7 marks]
3. Using the entire sample, compute and report the VaR95 of your assigned investment
style and discuss this result relative to that of the market portfolio. [8 marks]
4.
a. Using the first 325 observations, construct an optimal mean-variance strategy
with a risk aversion coefficient of 5 that combines the market portfolio and
your assigned investment style. Then, evaluate this strategy in the following
325 observations. This is, with the optimal parameters that give the optimal
combination in the first half of the sample, compute the returns in the second
half of the sample of that same combination between the market returns and
the investment style. Then, compute and discuss the Sharpe ratio obtained
from this strategy, and those of your assigned investment style and the market
portfolio in isolation. [5 marks]
b. Using the entire sample, construct the optimal mean-variance combination
with a risk aversion coefficient of 5 between the market portfolio and your
assigned investment style. Then, compute the Sharpe ratio of the resulting
strategy (e.g. the combination between the market returns and your assigned
investment style) and discuss this result relative to the Sharpe ratio of the
market portfolio and your assigned investment style in isolation. [5 marks]
c. Briefly discuss the difference between the results in part a) and part b). [5
marks]
REMARK: When computing the optimal portfolio, you should make sure that
the portfolio weights across all assets add up to one. You can read in the
Appendix how an investment style is constructed.
5. Using the entire sample, regress the returns from the strategy that optimally combines
the market portfolio and your assigned investment style on the three factors of Fama
and French. These factors are: the market portfolio (Mkt-rf), the small minus big factor
(SMB), and the high minus low factor (HML). Report the slope coefficients and the tstats and discuss your results. See the appendix for an explanation of the Fama and
French factors. [15 marks]
6. Market impact costs: when putting X dollars in your style, suppose that you move
prices against you (i.e. buying moves prices up and selling moves prices down).
Imagine that prices move against you according with this formula:
!”! = l” × ( ‘ − 0.003) + l# × ‘#
where TCp represent the transaction costs from assigning X units to your style, and l”
and l# are the transaction cost coefficients (i.e. the higher l” and l# are, the more
expensive it is to trade your investment style). Now, suppose that these coefficients
take a value of 0.005 and 0.004, respectively. Construct and optimal mean-variance
strategy that takes into account market impact costs using the first half of the sample
with a risk aversion coefficient of 5. Then evaluate the resulting optimal combination
between the market and your investment style in the second half of the sample.
Explain how your results change in comparison with your analysis in section/part 4.
[20 marks]
7. Using the entire sample, look at the 20 worst market episodes (i.e. 20 periods with the
lowest returns) and discuss how your assigned investment style helps you to cope with
the poor market performance. [10 marks]
8. After the implementation: Imagine a situation where you have implemented your
strategy for 6 months and you have experienced a loss of 50%. Give reasonable
arguments for this bad performance that would convince your boss of not firing you.
Be brief in your explanation and to the point. [15 marks]
• REMARK: Each time that you use solver, write down the mathematical problem that
you are solving with solver. For instance, if you were to use solver to construct a VaR
portfolio, you would need to write down in your report the mathematical problem
that gives you the optimal VaR portfolio.
General guidelines:
• In your submission, you should send the written report and the Excel file that you have
used to make computations. You should upload the documents as separate files. It is
not allowed to upload all documents in a zip file. If you have any problem with
submitting your work, feel free to send the submission to Mykola via email.
• Plagiarism will be highly penalized. Plagiarized works will receive zero marks.
Moreover, this behavior will be reported to the department and the university.
• 15% of the report mark will be for report presentation and writing skills
• The report has to be written in a professional way, making a smooth transition from
question to question. Divide the tasks described above in sections and subsections of
your report. Each sentence has to make sense. I do not recommend to make extensive
explanations that do not add value to the report. Try to be concise and clear.
• Word count must not exceed 3,500 and should be reported on the cover page:
o Excluding cover page, tables, list of references, appendices but including
footnotes and in-text citations.
• Word processed, 12 point font (Times New Roman or Ariel) and 1.5 line spacing.
• Ace my homework – Write my essay – Harvard Style Referencing is recommended, but any other reasonable style is also
accepted.
• The deadline for all soft copies (i.e. electronic submission via the 22/23 AcF351b
Moodle site) is Monday 16th January 2023, 12:00.
• Late mark penalties will apply for late submissions.
Appendix:
What is an investment style?
An investment style can be seen as a factor. This is a source of risk exposure in financial
markets. When we talk about the market value weighted factor, we think of the returns on
the market portfolio, which is usually represented by an index. The co-movement of a stock
with the factor defines the risk profile of the asset. There are many different factors such as
the value or the size factors. These factors are normally constructed with firm attributes that
are related with the factor. For instance, the value factor is constructed by sorting stocks on
the basis of the ratio book value to market capitalization (i.e. book value/market value). Once
we sort the entire universe of stocks on the basis of this firm characteristic, we go long in a
portfolio of companies in the top decile of book to market, and go short in a portfolio of
companies in the bottom decile of book to market. The size of these portfolios is the same,
and therefore the factor is constructed as the returns of a zero cost portfolio. The same idea
applies for the size factor or other factors constructed on other specific firm characteristics.
Constructing an investment style
Consider at time t a given firm specific characteristic (e.g. value, size, momentum). In addition,
consider a universe of N stocks. Therefore, you have at time t N different stocks. You sort
those stocks using the specific firm characteristic that you are interested in. Once the sorting
is done, you construct an equally (or value) weighted portfolio with the top N/10 companies,
and another equally (or value) weighted portfolio with the bottom N/10 companies. Then you
look at the spread of expected returns offered by these two portfolios, and if this spread is
large enough, you go long in the portfolio with the largest expected return, and short in the
portfolio with the smallest expected return. The return of this long-short (zero cost) portfolio
corresponds with the factor.
What are the three Fama and French factors?
Eugene Fama and Kenneth French are the godfathers of style investing.1 Their three factor
model is an asset pricing model that explains both the time series and the cross-section of
stock returns. The first factor is the market portfolio. The second factor is named small-minusbig (SMB). This is an investment style that buys one dollar of an equally weighted portfolio of
small companies, and shorts one dollar of an equally weighted portfolio of big companies.
The third factor is named high-minus-low (HML). This is an investment style that buys one
dollar of an equally weighted portfolio of companies with high book value relative to their
market value, and shorts one dollar of an equally weighted portfolio of companies with low
book value relative to their market value.
1 Style investing is more commonly known as factor investing. Investable factor are what we know as
investment styles, which are constructed as explained in the Appendix.

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