Problem 1: Portfolio Optimisation and Factor Investing

Working in a large investment fund as a quantitative analyst, your daily task involves tracking portfolio

weights of different factor trading strategies that your fund invests in. The factor trading strategies your

fund is implementing are value (HML - high book to market firms minus low book to market firms),

size (SMB - Small firms minus big firms), momentum (UMD - past winners minus losers), operating

profitability (RMW - Robust firms minus weak firms) and investments (CMA - Conservative firms

minus Aggressive firms).

The monthly factor returns for 4 trading strategies from July 1963 to December 2020 can be found in

equity_strategies.csv placed under the Assignment folder.

Using the provided strategy returns, complete the following tasks:

1) Report the annualised average return and annualised volatility for each strategy. (4 marks)

2) Report the correlation and covariance matrix. (6 marks)

3) Construct an efficient portfolio EP with a target return of 3%. Report and interpret the weights

on each asset. What is the portfolio risk? You need to use and report the Excel Solver steps for

this task. (5 marks)

4) Construct the global minimum-variance portfolio GMV. Report the weights on each asset and

the portfolio risk and return. You need to use and report Excel Solver for this task. (5 marks)

5) Your fund caters to multiple investors with different risk preferences. Using Excel Solver to

complete the following tasks:

a. Low risk portfolio A for the group of investors who requires 2% rate of return. What is

the portfolio risk? Is this portfolio efficient/inefficient? Explain why this portfolio is

efficient/inefficient. (6 marks)

b. Medium risk portfolio B for the group of investors who targets return at 4%. What is

the portfolio risk? What are the weights on individual trading strategies? (5 marks)

c. High risk portfolio C for the group of investors who requires 10% return. What is the

portfolio risk? What are the weights on individual trading strategies? (5 marks)

d. Repeat 5.a, 5.b, and 5.c but now impose short sale constraints. Note that Excel Solver

might not be able to find a solution. (6 marks)

e. Based on the results from 5a, 5b, 5c and 5d, do you agree/disagree that the mean

variance optimisation is a reliable framework? Why? Your answers should tie with

theories and practical implementation above. (5 marks)

6) Now you construct the portfolios in 5a, 5b, and 5c, using EP and GMV portfolios constructed

in Parts 3 and 4.

a. What is the covariance between EP and GMV? (5 marks)

Hint: It depends on the weights on each asset put in to construct each portfolios, risk of

each asset, and covariances.

b. What are the weights on EP and GMV to achieve portfolios A, B and, C? Are your

portfolios created by combining EP and GMV similar to those in 5a, 5b, and 5c in terms

of risk and return? (5 marks)

c. What you just did in Part 6 is simple and in fact you can span the entire efficient frontier

using the two efficient portfolios. This mathematical fact was thought to be an insult to

investment professionals. Explain conceptually why? (3 marks)