Evaluating Monthly Returns for Two Funds
Question: Consider the data on returns and log returns
for two Vanguard funds (VPU and VFINX).
VPU is a utility fund and VFINX is a fund that specializes in the
S&P 500.
·
Using this data, find the mean, standard
deviation, skew of the returns and typical percentile statistics for both
monthly return measures for the two funds.
What do these statistics tell us?
·
Conduct a formal hypothesis test on the equality
of the variances and the equality of the means.
Can we reject the hypothesis of equal variances or equal means?
A previous post at my finance blog found that placing
$200,000 in VPU in 2004 was much more lucrative than placing $200,000 in
VFINX.
The results from the hypothesis tests asked for in the
second bullet appear to contradict the results in the previous blog.
Please explain this contradiction?
Analysis: Statistics on return and log return for the
two funds are presented below.
Statistics on Return and
Log Return for VFINX and VPU


Return VFINX

Return VFU

Log Return VFINX

Log Return VPU


Average

0.0066

0.0077

0.0025

0.0030

Standard Deviation

0.0408

0.0374

0.0180

0.0165

Skew

0.7695

1.1172

0.9999

1.2546

Min

0.1679

0.1287

0.0798

0.0598

5th

0.0701

0.0629

0.0316

0.0282

25th

0.0159

0.0072

0.0069

0.0031

50th

0.0126

0.0132

0.0054

0.0057

75th

0.0316

0.0322

0.0135

0.0138

95th

0.0664

0.0544

0.0279

0.0230

Max

0.1091

0.0837

0.0450

0.0349

N

142

142

142

142

2004 to 2015 Time Period
Some observations:
·
Average returns from VPU (the utility fund) are
larger than average returns from VFINX (the S&P fund.)
·
For both funds the standard deviation of returns
are much larger than average returns.
·
The skew of all return variables is
negative. The median is larger than the
mean in all cases. The left tail is
larger than the right tail. The
distribution is not symmetric. Also, the
most negative return is larger in absolute value than the most positive
value.
The hypothesis tests for the equality of variances and for
the equality of means are presented below.
Hypothesis Test Results
on Differences In Return for VPU and VFINX
(Statistics below are
pvalues.)


Return

Log Return


F test for equal
variances

0.2941

0.3092

Paired Ttest on Means

0.7242

0.6951

TTest assuming equal
variances

0.8124

0.7942

Observations:
·
The hypothesis of equal variance of returns for
the two funds cannot be rejected at any traditional level of significance.
·
The variance of equal mean returns for the two
funds cannot be rejected at any traditional level of significance. (Note that both the paired t test and the
traditional difference in mean tests are shown.
Neither test rejects the hypothesis of equal means.)
A Contradiction: The previous blog compared the average dollar
balance of investing $200,000 in VPU in 2004 to the average dollar balance of
investing $200,000 in VFINX.
Below is one of the key results of that exercise.
Average and Standard
Deviation of Two $200,000 Investments


VFINX

VPU


Average Value

$277,128

$365,072

STD value

$83,119

$102,819

These results indicate the investor in VPU likely made a lot
more money than the investor in VFINX.
However, the statistical tests presented here fail to reject
the hypothesis that mean returns are indentical.
How can this be?
Observations:
Note the variability of returns for both funds is
large. The lack of a significant
difference between the returns on the two funds is largely due to the high
variability in monthly returns.
The variability of monthly returns for VPU was especially
sharp in 2015 when VPU fell quite a bit.
VPU did outperform VFINX in the overall 2004 to 2015 time period but not
so in many months, especially recent months.
Note also that the arithmetic mean of returns is a flawed
statistic. Consider a stock price
initially at $50. It falls 50% the first
period and rises 100% the next period.
The average return for the stock is (50+100)/25 or 25%. However,
the stock price is back to $50 and the gain for the entire period is 0%.
Another stock that had 0% movements in both periods would
have an average return of 0% and the same gain (0%) as the stock that went down
and went back up.
Note neither the return variable nor the log return
variables are normally distributed. Both variables for both funds have a negative
skew. Nonparametric tests are more
appropriate than parametric tests when data is not symmetric.
The next blog on this topic will present results of a
nonparametric test.
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