Predictive Models of Realized Variance Incorporating Sector and Market Variance

1. Predictive Models of Realized Variance Incorporating Sector and Market Variance Haoming Wang April 16th 2008

2. Motivation • We’ve seen that the realized variance of equities exhibits correlation with lagged daily, weekly, and monthly realized variance. • We know that equity returns are also correlated with market returns, what kind of correlation do equity variance levels have with market variance levels? • Further, it’s intuitive that the returns of individual equities are also correlated with the returns of companies in their own sector. What kind of correlation should we see with sector volatilities? • Examined regressions for pharmaceuticals and banks in the S&P 100.

3. Pharmaceutical Stocks • There are five companies in the S&P 100 classified as in the “Major Drugs” sector. • ABT – Abbott Labs • BMY – Bristol Myers Squibb • JNJ – Johnson & Johnson • MRK – Merck • PFE - Pfizer

4. Bank Stocks • Six companies in S&P 100 classified as money center banks. • BAC – Bank of America • BK – Bank of New York-Mellon • C – Citigroup • JPM – JP Morgan Chase • USB – US Bancorp • WFC – Wells Fargo

5. Motivation for Stock Choices • Want to choose two industries that were pretty different to see if my results weren’t just unique to the pharmaceuticals industry. • The bank industry and the pharmaceuticals industry seemed pretty different from each other. • Five pharmaceutical stocks have an average beta factor of 0.652. • Six bank stocks have an average beta factor of 0.945.

6. Background Mathematics • Realized variation (where rt,j is the log-return): • Realized sector variation: An average of daily realized variation for same sector stocks in the S&P 100 excluding whichever stock is being regressed. • Realized sector portfolio variance: average of daily stock prices is used as the price of the portfolio.

7. Background Mathematics • Realized variation (where rt,j is the log-return): • Realized semi-variance (where 1 is the indicator function that the return is negative) :

9. Background Mathematics • Further, according to BNKS, the realized semi-variance converges to half the bipower variation plus negative squared jumps, or:

10. HAR-RV Model • The multi-period normalized realized variation is defined as the average of one-period measures, or: • The daily hetereogeneous autoregressive realized variance (HAR-RV) model of Corsi (2003) is used with daily, weekly, and monthly periods:

11. HAR-RV Model • The above regression is for day ahead predictions, also looked at week-ahead and month-ahead regressions. • Finally, I looked log-log regressions, where the natural log of realized variance is regressed on the natural log of the regressors. • This gives an intuitive meaning: a 1% change in the regressors implies a β% change in the dependent variable.

12. Extended HAR-RV model • Added sector variance for daily, weekly, and monthly periods. • For example, the RV of PFE would be regressed on the average RV of ABT, BMY, JNJ, and MRK. • What happens when PFE RV is also included in the average? • The standard errors and R-squared remain the same; however, the interpretation is not as clear. • Further, the interpretation is trickier since the stock coefficient is incorporated into the sector coefficient. • Also included the RV of the S&P 500.

13. Sector As A Portfolio? • Investigated constructing a portfolio of the industry. • Across the board there’s higher standard errors and lower R-squared. • I decided to stick with using an average of the sector’s realized variance.

14. Data Range • Data for all 11 stocks and the S&P were gathered from 1997 to 2007. • 2606 trading days worth of observations for pharmaceuticals stocks. • 2519 trading days worth of observations for the bank stocks.

15. Results • We’ll focus on two companies: Johnson & Johnson (JNJ) and JP Morgan (JPM), one for each sector. • I’ve picked these two out because they show the greatest benefit from the inclusion of more regressors, but I’ll include averages as well.

18. Analysis • We see marked increases in the explanatory power of the regression model, with the greatest improvement in the weekly predictions. • This makes intuitive sense: it should be easier to predict smoothed out variance measures. • This is in line with the level of the R-squared: we have the highest level of R-squared in the weekly predictive model. • Seems intuitive that the model with the highest explanatory power should also see the most improvement.

19. Analysis • In the pharmaceuticals sector, we see an average increase in R-squared of 5.12% for daily predictions, 5.80% for weekly predictions, and 6.41% for monthly predictions. • In the banking sector, we see an average increase in R-squared of 6.24% in daily predictions, of 6.83% for weekly predictions, and 3.41% for monthly predictions.

23. Analysis • In all regressions we see that the lagged daily sector variance is strongly significant. • There are clear sector effects at work, although only the previous day sector RV seems to play a consistently statistically significant role. • However, sector regressors are jointly significant for all time horizons. • While the p-values for the sector regressors remains at around the same level, it’s interesting to note that for JNJ the S&P regressors become much more significant as the horizon increases, perhaps suggesting that market effects play out in a longer time-frame for JNJ.

24. Analysis • The S&P coefficients are harder to interpret: there are no clear patterns in significance, although daily and monthly S&P regressors seem to be significant for JNJ, however, there is still the weirdness of negative coefficients. • JPM is especially strange because the sign of the monthly S&P lag flips signs. • From looking at semi-variances, this seems to be driven by high upward semi-variance. • However, we do have a pretty clear pattern: the previous day’s sector RV does seem to be an important driver of realized variance.

25. Analysis • Increasing the time horizon also doesn’t seem to solve the problem of the negative monthly coefficients. • I’ll talk about this later, but what effect does the time-frame have on the model’s coefficients? • I plan on investigate this further for next semester.

26. Analysis • It’s also interesting to note that even with the inclusion of the S&P and sector, we still see the same-stock daily lags decreasing in importance as the time horizon increases and the monthly lags increasing in importance. • Further, the weekly lag seems to be relatively unimportant. It seems that the most important drivers are the short term and the long term. • For JNJ we see some intuitive results for the sector and the stock: the lagged daily effects grow weaker as the prediction horizon increases and the monthly effects are stronger.

39. Analysis • It’s interesting to note the patterns in RV predictions. • Regression models with sector and S&P realized variance seem to have much greater ability to exhibit spikes in realized variance. • This makes intuitive sense: it’s likely that events which cause spikes in volatility for the market or the sector to be associated with spikes in a specific stock.

40. Analysis • It’s also interesting to note trends in realized variance over this data set. The first graph is from about 1999-2001 and the second graph is from 2006-2008. • Day ahead predictions seem close for both time periods. • However, for the early part of 2006-2008 (pre-credit crisis) the model consistently over-predicts week ahead and month ahead realized variance. • It seems that 2006-2007 was a period of below average volatility in the market. • It would be interesting to see how the model changes for different periods of data corresponding to different volatility regimes.

41. Extensions For Next Semester • Breaking up realized variance into continuous and jump components for both the sector and stock:

42. Extensions For Next Semester • Including the sector and S&P realized variances seems to increase the prediction of spikes in realized variance, it would be interesting to see how the inclusion of jumps as regressors changes these dynamics. • Further investigation using semi-variance of the negative coefficient in monthly lags.