Thursday, June 16, 2016

Comparing proportion of down streaks for two funds

Stock Decline Proportions

Question:   The table below has information on the number of times a large-cap and a small-cap fund declines in a 10-day period.   The data for this problem consists of 1370 overlapping ten-day periods from January 2011 to mid June 2013.

What proportion of times did each fund decline 5 or more times in a 10-day period?   Are these proportions significantly different for a two-tailed test at a 0.05 level of significance?



Number of Declines in a Ten-Day Period
Large Cap Fund
Large Cap
 Fund
Small-Cap
Fund
0
1
2
1
20
25
2
103
87
3
236
216
4
343
313
5
321
333
6
226
256
7
96
104
8
22
31
9
2
3
10
0
0
Total
1370
1370

Analysis:


First, calculate the proportion of decline days 5 and over and the complement decline days fewer than 5.



Daily Declines in Ten Day Periods
% Declines in Ten Day Periods
Large Cap Fund
Small-Cap Fund
<5 Declines
703
643
0.51314
0.46934
>=5 Declines
667
727
0.48686
0.53066
1370
1370
1
1


Second, calculate the t-statistic. 

I like to calculate the standard error for the t-statistic without pooling data to get a common proportion for >=5 declines over the two funds.) 


t= ((0.48686-0.53066)/ ((0.48686*0.51314/1370) + (0.53066*0.46934)/1370))0.5


This is equal to t= -2.30.

Third, compare the area to the left of this t-statistic to 0.025, the size of the critical region defined by our test stipulations.

Here is a useful normal distribution table on the web.


The area to the left of -2.3 is 0.0107.  

This is small than 0.025.


We reject the null hypothesis that the proportion of 5 or more decline days out of 10 is identical for the two funds.


The likelihood of a large number of decline days seems large for the small-fund ETF.


A Finance Footnote:  I have numbers for a third fund the total stock market funds, which includes large-cap, mid-cap and small-cap stocks.   This fund realized 706 occurrences with fewer than 5 of 10 down days and 664 with more than 5 declines in 10 days.   The number of large decline periods is slightly smaller for the total stock market fund than for the large-cap fund.  But this difference is not significant.


Authors Note:  I am putting lists of problems on probability and statistics on the blog statistical resources. 

The probability problems in this post will involve the binomial distribtution.







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