Problem Set 3
You may work with other students. The maximum number of students per group is two. However, you can work on your own. Be sure to indicate with whom you have worked in your submission.
There is a penalty for late submissions: will be subtracted from the total mark for every additional day after the deadline. If you submit it after Dec 15, 2024, you will get a zero on this homework assignment.
Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The cross-section of volatility and expected returns. Journal of Finance, 61(1), 259-299.
In this problem set, you will examine the pricing of volatility risk in the cross-section of stock returns, following the Journal of Finance paper Ang, Hodrick, Xing, and Zhang (2006) (thereafter, AHXZ (2006)). Specifically, we ask the following questions: Do the stocks with larger exposures to the volatility risk earn higher or lower average returns?
To answer this question, we first need to find a measure of volatility exposure. Following AHXZ (2006), we consider the VIX index. The VIX index is constructed so that it represents the implied volatility of a synthetic at-the-money option contract on the S&P100 index that has a maturity of 1 month. It is constructed from eight S&P100 index puts and calls and takes into account the American features of the options contracts, discrete cash dividends, and microstructure frictions such as bid-ask spreads.
Part I. Main Findings of AHXZ (2006)
In this part, you need to summarise the main findings in AHXZ (2006). Please list the two findings that you think are the most important (there are more than two, but you do not have to list all of them). For each key finding, please provide an economic explanation of the empirical phenomenon.
The monthly and daily individual stock data come from CRSP, accounting data come from COMPUSTAT, and data on the CBOE implied volatility index, VIX, come from the FRED St Louis. To be clear, AHXZ (2006) use the SP100-based implied volatility index, which has a ticker of VXO, for all tests reported in this paper. You can download the Fama-French three factors (market, size, and value factors) from Ken French's website. I provide some useful links to several datasets at the end of this document.
Your first task is to download all the data and load the datasets using pandas . After that, you need to report (1) which datasets you use in this problem set and why you need them, (2) how you preprocess the data (e.g., dropping samples based on some requirements, handling missing data, merging datasets, etc.), and (3) how many firms per year your final sample has in the panel data of stock returns.
To measure the sensitivity to aggregate volatility innovations, you are required to run the following regression:
You need to run the above regression with daily data for each stock per month.
At the end of each month, you sort stocks into quintiles based on the value of the realized coefficients over the past month. Firms in quintile 1 have the lowest coefficients, while firms in quintile 5 have the highest loadings. Within each quintile portfolio, we value-weight the stocks. We link the returns across time to form one series of post-ranking returns for each quintile portfolio. In this portfolio sorting step, you should use the CRSP monthly stock return data.
Your task is to replicate the empirical results in Table I of AHXZ (2006). Please only replicate the numbers in the following attached table.
The first two columns report the mean and standard deviation of the monthly total, notexcess, simple returns.
The column labelled % Mkt share shows the percentage of market cap for all the stocks in each quintile.
The sample period in AHXZ (2006) is from January 1986 to December 2000. However, I require you to conduct the same data analysis in the out-of-sample, January 2001 to December 2020.
Caveat: It is impossible to get exactly the same numbers as in the original paper.
Some useful links:
- VXO/VIX index: https://wrds-www.wharton.upenn.edu/pages/get-data/cboe-indexes/cboe-indexes-1/cboe-indexes/ or https://fred.stlouisfed.org/series/VXOCLS
- Ken French's library: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
- CRSP Stock / Security Files: https://wrds-www.wharton.upenn.edu/pages/get-data/center-research-security-prices-crsp/annual-update/stock-security-files/
To save you time, I put the daily and monthly stock return data in Dropbox.
However, you need to download the VXO index and Fama-French factors on your own, using the above links.
You need to submit two documents:
- A PDF file that contains your explanation, such as the main findings, economic explanations, execution details, etc. Please keep your document as concise a possible (no more than five pages).
- Python codes (Jupyter Notebook, .py file, etc.) that show the details of your data work (Please add as many comments as you can).
Your grade is determined by the accuracy of your solutions, explanations of each data analysis step, and your interpretation of the empirical findings.
Please do NOT submit the datasets.
Finally, if you find anything unclear, please read the JF paper, AHXZ (2006), carefully.