Assignment 2 (due October 20)

Part I: Analyzing the Mean-Variance Efficient Frontier

The first exercise asks you to analyze the monthly returns data on the 12 industry portfolios in the industry returns.xlsx file in the Assignment 2 folder on Canvas. The data were downloaded from Ken French’s website and cover the period 1926:07 – 2025:07. You can assume that the risk-free rate is 4% per annum or 0.333% per month. All returns are in percent per month.

1. Using returns on the 12 industry portfolios in columns B-M, construct and plot the efficient frontier for a reasonable range of expected return values (e.g., 0 - 1.5% per month) [Hint: you can do this using the formulas in the lecture handout which are coded up in the Assignment_2_sample_code.py python code. Be careful that all return series, the risk-free rate and the target mean return are on the same scale, i.e., in decimals and per month. It is common practice to convert percentage returns into decimals by dividing all stock return series, the risk-free rate and the target mean return by 100.]

2. Report the weights of the tangency portfolio (12 by 1 vector). How large is this portfolio’s Sharpe ratio?

3. Suppose we constrain the weights on each of the industries to lie between zero and 20%. Compute the optimal weights and the associated Sharpe ratio of the portfolio that is subject to these limits on the portfolio weights. (Again, you can use the python code to get started).

4. Using data up to 2009:12, compute the portfolio weights on the 12 industries which maximize the Sharpe ratio for this data sample. Then, using these weights, compute the actual Sharpe ratio on returns data for the remaining sample, 2010:01-2025:07. Also compute the Sharpe ratio for 2010:01-2025:07 for an equal-weighted portfolio that puts a weight of 1/12 in each industry. How do the two strategies (optimal Sharpe ratio weights vs. equal weights) compare in terms of out-of-sample performance?

Part II: Back-Testing the Momentum Effect with Industry Portfolios

Momentum in individual stocks’ return performance has been linked to industry effects, the notion that certain industries’ returns go through “hot” and “cold” spells. We will conduct a back test to see how a momentum investment strategy of going long in “Winner” industries (industries with high past returns), and shorting “Loser” industries (industries with poor past returns), would have performed if implemented in “real time”.

To address this question, we will again use the monthly returns data for the 12 industry portfolios in the Canvas file industry_returns.xlsx.

You are asked to compute an estimate of the momentum in month t as the average return, within a given industry, from month t-12 through month t-2 (i.e., using 11 monthly observations). We skip one month to account for liquidity and price effects. This estimate needs to be updated each month and so industry rankings may change frequently.

Using the ‘skip one month’ average returns over the past 12 months, rank each of the 12 industries. Then form. a Winner portfolio of the top ranked industry (the industry with the highest past mean return) and a Loser portfolio of the bottom ranked industry (lowest past mean return). Repeat this procedure every month in the sample, starting in 1950:01.

1. Compute the average returns of the Winner (W) and Loser (L) portfolios as well as that of the WML (winner minus loser) portfolio that goes long in winners and shorts losers. Also compute the standard deviation of the W, L, WML portfolios. What do you conclude?

2. Plot the time-series of rolling 3- and 10-year average returns on the industry Winner minus Loser (WML) portfolio. Comment on what you see, in particular, has the performance of the WML portfolio deteriorated over time?

3. Which industries are most often included in the Winner portfolio?

4. Which industries are most often included in the Loser portfolio?

5. How large is the ‘turnover’ on the WML portfolio, i.e., how often are different industries rotated in and out of the WML portfolio?

6. Discuss if the WML portfolio is a desirable portfolio that can be implemented in practice. Use different measures of its risk such as volatility, skew and maximum drawdown as well as the expected return to support your conclusion.

7. Regress the WML returns on an intercept and the Mkt-RF, SMB and HML Fama-French risk factors in columns O, P, and Q to obtain the WML portfolio’s alpha (risk-adjusted abnormal return). How large is the alpha and is it statistically significant? Is the alpha coming from the Winner or Loser portfolio returns or both?

Part III: Replicating HML via double sorts

Using the financial ratios suite in WRDS, download data on individual firms’ book to market ratios. Also download data on the associated stock returns. To do this, refer to the additional document “Instructions for WRDS Access”. With all this data in hand, merge the two dataframes so you have a single dataframe. with columns consisting of returns, accrual ratio, and market cap for each firm (identified by PERMNO) across time.

For each quarter (at the end of the quarter) in the dataset, form. a “high” portfolio of the top 50% of stocks with the highest market capitalization and a “low” portfolio comprising the 50% of stocks with the lowest market capitalization. Then sort firms on their book to market ratios and form. a “high-value” portfolio consisting of the firms with the highest 30% of book to market ratios within the “high” market capitalization stocks. Also, form. a portfolio of “high-growth” where you take the bottom 30% of book to market ratios within the “high” market capitalization stocks. Repeat this but with the “low” market capitalization stocks to get “low-value” and “low-growth”. In total you now have four groupings of stocks. Note that which firms are in these four groups changes each quarter.

With these grouped firms, form. portfolio returns by using value weights calculated at the end of the quarter for the next three months. Portfolio weights and portfolio composition should be updated every quarter based on firms’ market cap and book to market ratio, respectively. Returns must be measured at the monthly frequency. Do this for the whole sample specified in the “Instructions for WRDS Access” document (1970-01 to 2024-12), and answer the following questions:

1. What are the average returns of these portfolios?

2. What is the standard deviation of returns of these portfolios?

3. Construct a zero dollar portfolio by going long on  “low-value” and short on “high-growth”. Compute and report the average and standard deviation of returns on this portfolio.

4. Download the Fama French 3-factors from Ken French’s website. What is the correlation between your zero-dollar factor and the HML factor you downloaded?

5. Plot the return of one dollar invested in your zero-dollar portfolio against another dollar invested in the Fama French HML factor. Are there any significant deviations from one another? What might cause such deviations? Consider the differences between the way you constructed the factor versus how Fama French construct it.