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 25^{th} Percentile, and the 75^{th} Percentile)
for profits from the two funds over the two time periods. I also have the microdata 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 25^{th} and 75^{th} percentile of profits for all funds. How close are estimated 25^{th} and 75^{th} percentiles to the actual 25^{th} and 75^{th} 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 25^{th} and 75^{th} percentile:
The estimated value of the 25^{th} and 75^{th}
percentile under the assumption of normality is
Estimated Percentile
= mean + or  0.674 % STD
In the above, Z_{0.25}= 0.674 and Z_{0.75}
=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%

pvalue from sktest

0.57

0.17

0

0

Observations:
·
The deviations between actual and expected 25^{th}
percentile profit outcomes and actual and expected 75^{th} 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 pvalues 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 todo 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.
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