## Friday, January 29, 2016

### Are Investment Fund Profits Normally Distributed?

Are Investment Fund Profits Normally Distributed?

In a previous post I calculated the average and standard deviation of investment profits from six Vanguard funds.  I generate statistics on fund profitability for funds purchased in 2007 and sold in 2009.   I also generate fund profitability numbers for funds purchased in 2001 and sold in 2014.

For a discussion of the generation of these profitability statistics go to the following post.

The post today looks at whether the fund profit numbers generated in the previous posts are normally distributed.

The two funds that we are considering here are VFIAX (a fund that invests in the S&P 500) and VBLTX (a fund that specializes in investment grade and U.S. government bonds.)

Question:  Below are statistics (mean, Std, Skewness, Kurtosis, the 25th Percentile, and the 75th Percentile) for profits from the two funds over the two time periods.   I also have the micro-data on fund profit outcomes.

•  Discuss what one might learn by simply looking at the summary statistics on profits for the four funds?
•  Under the assumption that the data is normally distributed create estimates of the 25th and 75th percentile of profits for all funds.   How close are estimated 25th and 75th percentiles to the actual 25th and 75th percentile for each fund?
• What profit numbers appear to be normally distributed based on the above comparison
• Do formal tests for normality support your conclusions on what and when fund profits are normally distributed?

 Descriptive Statistics for Profits from a Stock and  Bond Fund (2007 to 2009) Time Period 2001 to 2014 2007 to 2009 Fund VFIAX VBLTX VFIAX VBLTX Mean 11128 15345 -3254 1166 Std 1823 1364 911 619 Skewness 0.17 -0.06 -0.17 0.07 Kurtosis 2.7 2.4 1.9 1.8 25th Percentile 9753 14480 -4018 663 75th Percentile 12368 16293 -2477 1728

Answer:  Here are some general observations on mean, standard deviation, skewness, and Kurtosis of returns.

• Mean return of bond fund is greater than mean return of stock funds for 144 outcomes over both time periods.  Standard deviation of returns from bond fund is lower than standard deviation of returns from stock funds for all 144 outcomes over both time periods.  The finding of higher returns on the stock fund than the bond fund is a bit unusual; although, easily explained by what happened to interest rates over these time periods.
• The reported skewness numbers are fairly small and my initial impression is that these distributions are fairly symmetric.
•  Kurtosis near 3 suggests the distribution has a shape close to normally distributed. The kurtosis figures for the 2007 to 2009 time period suggest that returns for this narrow period are not normally distributed.

Discussion of actual and estimated 25th and 75th percentile:

The estimated value of the 25th and 75th percentile under the assumption of normality is

Estimated Percentile = mean  + or -  0.674 % STD

In the above, Z0.25= -0.674 and Z0.75 =0.674

Below we have data on actual, estimated and percentage difference actual versus estimated percentiles.

 Considering the Normality of Fund Profit Data 2001 to 2014 2007 to 2009 Fund VFIAX VBLTX VFIAX VBLTX 25th Percentile 9753 14480 -4018 663 75th Percentile 12368 16293 -2477 1728 estimate of 25th percentile based on normal distribution 9898 14425 -3868 748 Estimate of 75th Percentile based on Normal Distribution 12358 16265 -2640 1584 Percent difference actual 25th and estimated 25th -1.5% 0.4% 3.7% -12.9% Percent difference actual 75th and estimated 75th 0.1% 0.2% -6.6% 8.4% p-value from sktest 0.57 0.17 0 0

Observations:

·      The deviations between actual and expected 25th percentile profit outcomes and actual and expected 75th percentile profit outcomes appear really small for both funds over the 2001 to 2014 time period.   I suspect that profits from both funds are normally distributed for this time period.

·      The story is more complex for the 2007 to 2009 time period.   There are substantial differences between the actual distribution and the one expected under normality for both funds in this time period.   The stock fund profits appear bunched more tightly than what one would expect under the normal distribution.  The bond fund distribution appears wider than one would expect under the normal distribution.

Results from a formal test of normality:

I use SKTEST to test for normality.  SKTEST is a tool in the STATA package.  It combines a test for normality based on skewness with a test for normality based on kurtosis.   Some details of this test procedure can be found at the link below.

The p-values from SKTEST indicate that the null hypothesis of normally distributed profits should be rejected for both funds in the 2007 to 2009 time period.   I cannot reject the null hypothesis of normally distributed profits for either fund over the 2001 to 2014 time period.

Concluding Thoughts:  One of the things on my to-do list is to read more about statistics like skewness and kurtosis and more about tests on distribution type.   Just looking at the skewness number or the kurtosis number provides limited information.

For instance, a distribution can be negatively skewed because of one outlier on the left with a large negative value or because there are a lot of observations in the lower tail or perhaps because of some sort of truncation at the right tail.   Similarly, low or high kurtosis could have multiple causes.  Perhaps graphs are a better approach to this problem.

Authors Note:

Some people have found my book Statistical Applications of Baseball useful.  It has a lot of problems many of which have been used in class rooms.   Please consider it.