Saturday, July 21, 2018

A statistician’s thoughts on financial return statistics for overlapping holding periods



This post looks at the validity of statistical tests when observations are not independent. The specific application involves financial return data with overlapping holding periods.

Question:   The first chart below contains financial return data on large cap value and growth stock for 16 holding periods.   The second chart below contains financial return data for small cap value and growth stocks for the same 16 holding periods.

What is the difference in average returns for growth and value stocks for large cap and small-cap stocks?

Conduct paired t-tests for the hypothesis that there is no difference between mean return of growth and value stocks for small-cap and large-cap stocks using the return information from the 16 overlapping holding periods.

Why is the use of this test on these samples created from overlapping holding periods problematic?

Could an average of 16 holding periods be used in a cross-sectional study of ETF financial returns?  Would use of such an average in a cross-sectional study be worse or better than a cross-sectional study using a single holding period?

Data:  

Returns on Large-Cap Value and Growth Funds
Obs No.
Purchase Date
Sale Date
VTV Value fund
VUG Growth Fund
 VUG-VTV
1
7/1/13
7/1/17
11.0%
13.6%
2.6%
2
10/1/13
7/1/17
10.9%
12.4%
1.5%
3
1/1/14
7/1/17
11.3%
12.5%
1.2%
4
4/1/14
7/1/17
9.7%
12.2%
2.5%
5
7/1/13
10/1/17
11.5%
14.1%
2.6%
6
10/1/13
10/1/17
11.4%
13.0%
1.6%
7
1/1/14
10/1/17
11.8%
13.2%
1.4%
8
4/1/14
10/1/17
10.3%
12.9%
2.6%
9
7/1/13
1/1/18
13.0%
15.7%
2.7%
10
10/1/13
1/1/18
13.1%
14.7%
1.6%
11
1/1/14
1/1/18
13.5%
15.0%
1.5%
12
4/1/14
1/1/18
12.2%
14.9%
2.7%
13
7/1/13
4/1/18
10.8%
13.5%
2.7%
14
10/1/13
4/1/18
10.7%
12.6%
1.9%
15
1/1/14
4/1/18
11.0%
12.7%
1.7%
16
4/1/14
4/1/18
9.6%
12.4%
2.8%


Returns on Small-Cap Value and Growth Funds
Obs No.
Purchase Date
Sale Date
VBR Value
VBK  Growth
VBK-VBR
1
7/1/13
7/1/17
10.8%
8.8%
-2.0%
2
10/1/13
7/1/17
10.2%
7.5%
-2.7%
3
1/1/14
7/1/17
10.1%
7.0%
-3.1%
4
4/1/14
7/1/17
9.2%
7.9%
-1.3%
5
7/1/13
10/1/17
11.2%
9.7%
-1.5%
6
10/1/13
10/1/17
10.7%
8.6%
-2.1%
7
1/1/14
10/1/17
10.6%
8.1%
-2.5%
8
4/1/14
10/1/17
9.8%
9.1%
-0.7%
9
7/1/13
1/1/18
11.8%
10.8%
-1.0%
10
10/1/13
1/1/18
11.3%
9.8%
-1.5%
11
1/1/14
1/1/18
11.3%
9.4%
-1.9%
12
4/1/14
1/1/18
10.5%
10.4%
-0.1%
13
7/1/13
4/1/18
10.3%
9.9%
-0.4%
14
10/1/13
4/1/18
9.7%
8.8%
-0.9%
15
1/1/14
4/1/18
9.6%
8.5%
-1.1%
16
4/1/14
4/1/18
8.9%
9.3%
0.4%


Statistical Analysis:

Calculation: othe paired t-test statistic:  Just take the average and standard deviation of VUG-VTV for large-cap stocks and VBK-VBR for small-cap stocks.x

The t-statistics is the average divided by the standard error, which is the standard deviation divided by the square root of the sample size.   The sample size is 16.   The square root of the sample size is 4.

Results:   The observed averages and standard deviation between returns for growth and value stocks in both the large-cap and small-cap sector are presented below.


Difference Between Growth and Value Stocks
Statistic
Large Cap Stocks (VUG-VTV)
Small Cap Stocks (VBK-VBR)
average
2.11%
-1.40%
Standard Deviation
0.58%
0.96%
t-statistic
14.54
-5.83



In the large-cap sector, growth stocks outperformed value stocks

In the small cap-sector, value stocks outperformed growth stocks.


Discussion of validity of test:  The sixteen observations are not independent because the holding periods overlap.  Statistical tests can provide misleading results when the assumption of independent observations is violated.


Could an average of sixteen holding periods be used in a cross-sectional study of ETF financial returns?  Would use of such an average in a cross-sectional study be worse or better than a cross-sectional study using a single holding period?

Consider a cross-sectional study where the analyst wants to examine whether one type of fund has better returns than another type of returns. For example, the post below presents overlapping return statistics for 10 sector-specific ETFs and 6 broad-market ETFs.


Could we use this data to test for differences between performance of sector funds and broad funds?

The short answer is yes.   Components of the average overlap.  However, the assumption that returns from each sector are independent is not altered by the use of overlapping holding periods.

In my view, the use of a return measure based on multiple purchase and sale dates provides a more meaningful measure of return than a statistic based on a single purchase and single sale date.   The use of one holding period can provide unusual readings due to volatile stock market conditions.  

Final Note:   The difference in ou9comes growth minus value for large-cap versus small cap stocks is fascinating.   I need to follow up on this issue.





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