Problem Set 3

This is the final homework assignment, which accounts for 60% of your final grade. Unlike the previous problem sets, you are required to collect the data on your own and conduct data analysis based on your collected data.

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.

Deadline: Dec 5, 2024 (HK Time 11:59 PM).

There is a penalty for late submissions: 5% 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.

Reference

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.

Background

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.

Because the VIX index is highly serially correlated with a first-order autocorrelation of 0.94, we measure daily innovations in aggregate volatility by using daily changes in VIX, which we denote as ΔVIX.

There are three parts in this problem set.

PartI. 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.

Part II. Collecting Data

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.

Part III. Pricing Aggregate Volatility Shocks

To measure the sensitivity to aggregate volatility innovations, you are required to run the following regression:

rt(i) = β0(i) + βM(i)KT−RF ⋅ MKT − RFt + βΔ(i)VIX ⋅ ΔVIXt + ϵt(i) ,

where:

.  MKT-RF is the daily market excess return,

·   ΔVIXt is the daily change in the VIX index, and

·   βM(i)KT and βΔ(i)VIX are firm i's loadings on market risk and aggregate volatility risk, respectively.

You need to run the above regression with daily data for each stock per month.

Specifically, for each month, you run the regression for all stocks on AMEX, NASDAQ, and the NYSE, with more than 17 daily observations and obtain the monthly estimates of βM(i)KT and βΔ(i)VIX . In this step, you will need to use the CRSP daily stock return data and also the market and VIX daily data.

At the end of each month, you sort stocks into quintiles based on the value of the realized βΔ(i)VIX coefficients over the past month. Firms in quintile 1 have the lowest coefficients, while firms in quintile 5 have the highest βΔ(i)VIX 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, not excess, simple returns.

.  The column labelled % Mkt share shows the percentage of market cap for all the stocks in each quintile.

.  The columns labelled size and B/M show the average log market capitalization and book-to-market ratio for firms within the portfolio (You do NOT need to replicate  these two columns).

.  The columns labelled “CAPM Alpha” and “ FF-3 Alpha” report the time-series alphas of these portfolios relative to the CAPM and to the FF-3 model, respectively.

·   The final column reports the pre-formation βΔ(i)VIX coefficients, which are computed at

the beginning of each month for each portfolio and are value-weighted.

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.

Does the long-short portfolio (marked as 5-1 above) have similar performance in the more  recent sample from January 2001 to December 2020? How do you interpret your findings?

Caveat: It is impossible to get exactly the same numbers as in the original paper.