Using the return data in the Validation set, construct another portfolio called “Combo.” This “Combo” portfolio invests wValw_{Val}wVal​ in the Max SR, and 1−wVal1 – w_{Val}1−wVal​ in the GMV

Exercise 1.1

Construct the following set of portfolios using return data in the Training set:

1. Equally weighted portfolio (EW)
2. Portfolio maximizing the Sharpe ratio (Max SR)
3. Portfolio minimizing the variance (GMVP)
4. Portfolio minimizing the variance subjected to the “no short sale” constraint (GMVP no short sale)

a. Compute the optimal portfolio weights for each of the four portfolios.
b. For each of the portfolios, calculate and interpret the following performance measures:

• Annualized average excess return
• Annualized volatility of excess returns
• Annualized Sharpe ratio
• Market beta
• Annualized alpha to the market
• Annualized alpha to the minimum variance portfolio
• Maximum drawdown
• Skewness of monthly excess returns
• Excess kurtosis of monthly excess returns

Exercise 1.2

Using the return data in the Validation set, construct another portfolio called “Combo.” This “Combo” portfolio invests wValw_{Val}wVal​ in the Max SR, and 1−wVal1 – w_{Val}1−wVal​ in the GMVP portfolios constructed in Exercise 1.1 to maximize Sharpe ratio in the Validation set.

Exercise 1.3

Using return data in the Testing set, perform the following tasks:

a. For the portfolios EW, Max SR, GMVP, GMVP no short sale constructed in Exercise 1.1, using the Training set and held fixed during the Testing period, and the Combo portfolio constructed in Exercise 1.2, calculate the following performance measures:

• Annualized average excess return
• Annualized volatility of excess returns
• Annualized Sharpe ratio
• Market beta
• Annualized alpha to the market
• Annualized alpha to the minimum variance portfolio
• Maximum drawdown
• Skewness of monthly excess returns
• Excess kurtosis of monthly excess returns

b. Discuss the difference in performance between GMVP and GMVP no short-sale, and also the difference in performance between Combo and Max SR. Give some reasonable explanations.
Compare the performance between the four portfolios in part (a), namely EW, Max SR, GMVP, GMVP (no short sale), Combo, and that of the Equally-weighted portfolio (EW), and that of the market (Mkt) portfolio.
Discuss the issues with portfolio optimization.

c. For each of the EW, Max SR, GMVP, GMVP (no short sale), Combo portfolios, run a multivariate regression of the portfolio excess return on the Mkt-RF, SMB, HML and Mom factors.
Compare multivariate alpha with the alpha from the univariate market regressions.
Discuss the difference in interpretation.

Exercise 1.4

Using return data in the Testing set and Validation set, perform the following tasks:

a. For each of the portfolios, namely, EW, Max SR, GMVP, GMVP (no short sale), Combo, and Mkt, and each month from January 2019 – December 2024, compute the annualized volatility as the realized standard deviation of excess returns over the past 12 months.
For the first year 2019, you need to use the excess returns from 2018 in the Validation set.
Compute the annualized volatilities of portfolios in the same graph and briefly comment on the time series patterns.

b. For each of the portfolios in part (a), construct, in turn, the return of the risk-managed strategy. Specifically, choose a constant xxx such that:

xt=3×MADD−DDtσtx_t = frac{3 times MADD – DD_t}{sigma_t}xt​=σt​3×MADD−DDt​​

where σtsigma_tσt​ is the current volatility, DDtDD_tDDt​ is the current drawdown, and MADD = 30% is the maximum acceptable drawdown.
Comment on whether or not the risk-managed strategy has worked.

c. For each of the five portfolios in part (a), compute the following performance measures:

• Annualized average excess return
• Annualized volatility of excess returns
• Annualized Sharpe ratio
• Market beta
• Annualized alpha to the market
• Annualized alpha to the minimum variance portfolio
• Maximum drawdown
• Skewness of monthly excess returns
• Excess kurtosis of monthly excess returns