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 ttest
for differences in means. But first I
consider whether the variances are identical.
The Fstatistic for the equality of variances is (0.0068)^{2}/(0.0058)^{2}
or 1.36. The pvalue for this
Fstatistic is 0.02.
Based on this test use the ttest for unequal
variances.
The test statistic is 4.87.
The pvalue 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 nonparametric 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 tstatistic of 6.12.
Again the pvalue 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.
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