## Friday, December 11, 2015

### Comparing returns from two Vanguard funds

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.

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 p-values.) Return Log Return F test for equal variances 0.2941 0.3092 Paired T-test on Means 0.7242 0.6951 T-Test 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.   Non-parametric tests are more appropriate than parametric tests when data is not symmetric.

The next blog on this topic will present results of a non-parametric test.