## Thursday, September 22, 2016

### Comparing Stock Price Volatility for Two ETFs

Comparing Stock Price Volatility for Two ETFs

Question:   The statistics below pertain to the daily stock price volatility of two stock ETFs offered by Vanguard – VV a large cap ETF and VB a small cap ETF.    Daily stock price volatility presented here is the difference between the high stock price of the day and the low stock price of the day as a percent of the low stock price.

Conduct a test of the hypothesis that the mean volatility for the two stock funds is identical compared to the alternative hypothesis that average volatility for the two funds differ

The Data:

 Daily Price Volatility for Two Stock ETFs VV Large Cap Stocks VB Small Cap Stocks average 0.0097 0.0129 Min 0.0029 0.0034 Max 0.0354 0.0504 Std 0.0058 0.0068 Count 182 182

The data covers daily observations over the period starting the beginning of January in 2016 to September 21, 2016.

Analysis:   I am going to conduct the standard t-test for differences in means.   But first I consider whether the variances are identical.   The F-statistic for the equality of variances is (0.0068)2/(0.0058)2 or 1.36.  The p-value for this F-statistic is 0.02.

Based on this test use the t-test for unequal variances.

The test statistic is 4.87.

The p-value for this test statistic is very small.

We reject the null hypothesis that the average daily volatility of the two ETFS are identical over this time period?

A Follow Up Question:  I went back looked at the data and calculated the skew of the two daily volatility measures.   I found

 Skew of Daily Volatility Measures January to September 2016 VV Large Cap Stocks VB Small Cap Stocks skew 1.63 1.98

Discuss what it means for these measures to be positively skewed and discuss the implications of the positive skew for the results of the hypothesis test presented above.

Conduct an alternative test that corrects for the issue created by the positive skew.

Discussion of positive skew:  The positive skew suggests that there are some large positive observations, which have a significant impact on the mean.  In other words, the data is not symmetric around the mean.

I am not surprise that the volatility measure is positively skewed.   On a volatile day there is no limit on the volatility index.   By construction the volatility index can never be less than zero.

Discussion of the implications of the positive skew on the hypothesis test:

The positive skew means the distribution is not normally distributed and the estimated probability of rejecting the null hypothesis is incorrect because we assumed normality.

There are two possible solutions.  The first used by most statisticians would be to run a non-parametric test, perhaps the Wilcoxon rank sum test.

The approach that I am going to use here is to take the logarithm of the data examine the data and and test the hypothesis on the log of the volatility indices.

 Analysis of Log of Volatility Measure Log (large Cap Index) Log Small Cap Index average -4.7838 -4.4597 Min -5.8300 -5.6696 Max -3.3411 -2.9878 Std 0.5404 0.4656 skew 0.30 0.28 Count 182 182

The data presented above suggests that the skew of the log of the daily price indices is much smaller than the skew of the untransformed data.

I test the null hypothesis that the logs of the volatility indices for the stock ETFs are identical.  I get a t-statistic of 6.12.

Again the p-value is very small and we reject the null hypothesis.

It would be interesting to examine the relative volatility of the two indices in a different period, perhaps in the financial crisis.