## Wednesday, July 17, 2019

### Comparing proportion of down streaks for two funds

This post starts teaches how to conduct a hypothesis test about proportions with data on stock declines 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.