## Thursday, December 17, 2015

### Statistical Tests of Identical Fund Returns

Question:   Consider data on daily fund returns over the 2004 to 2105 time period for FVPU and FVINX.   What are the mean and standard deviation of daily returns for the two funds over this time period?  Do paired versus unpaired tests reveal significant differences in the return of FVPU and FVINX?   Do non-parametric tests reveal evidence that the returns from the two funds differ in a significant way?

Description of Data: The data on daily returns for FPU and FVINX involve adjusted closing stock price for all days from 2/2/04 to 12/8/15.   The sample has 2985 observations.

The return variable is the ln(Stock Price at period t/Stock Price at t-1).

The mean and standard deviation of the stock prices are as presented below.

 Mean STD ret_ut 0.0003371 0.0002067 ret_sp 0.0002791 0.0002249

Observations:

·      The mean return is slightly higher for the utility stocks.

·      The standard deviation of returns for the utility stocks is slightly lower.

It is also worth looking at the correlation of daily returns between the two stock funds.

Observations:

·      The correlation between ret_sp and ret_ut is 0.7379.

This is pretty high but it may be the case that some diversification across funds is better than no diversification across funds, a topic for a future post.

Results from four statistical tests:  Are the difference in returns from the two funds statistically different from zero?

I address this issue with STATA software.  Four tests are considered – (1) the two- sample test with equal variances, (2) a paired mean test, (3) A Wilcoxon signed rank test on difference between two medians, and (4) a paired sign rank test.

The two-tailed p-values for the four tests are presented below.
 Test Two-Tailed p-value unparied t-test 0.8493 paired t-test 0.7119 Unpaired Wilcoxon Test 0.2982 Paired Sign Test 0.2001

Observations:

·      None of the four statistical tests reveal a significant difference between daily returns from the S&P fund and daily returns from the utility fund.
·      The p-values are lower for the non-parametric tests but still the evidence does not reveal significant differences in returns from the two securities.

Comment on this result:  Previous research found that the person who invested in the utility fund earned more money over the 2004 to 2015 time period than the person who purchased the S&P fund.

Profit opportunities may temporarily exist even when return differences are not significantly different from zero.

This result may not persist.  Returns from the utility fund started to fall in early 2015.  Also, both funds fell precipitously during the crisis of 2008 and diversification across funds would not have protected investors in that turbulent time period.   (Really the only thing that would have minimized risk in that period was a lot of cash.)

Additional Research:  I have at least two additional questions related to these two funds.   First, what are the potential gains from diversifying between the two funds?

Second, what is the expected return and standard deviation of return from purchasing funds some time in second quarter of 2004 and selling the stock sometime in the second quarter of 2015?   A method of measuring expected return and variability of return for stocks purchased and sold over particular time intervals strikes me as a useful return measure compared to more common return measures that pick a single purchase date and a single sale date.