Evaluating Fund Performance

Investment funds, both ETFs and mutual funds, are usually compared on the basis of returns of arbitrarily selected holding periods. Typically, the fund manager reports year-to-date returns and return for one, three, five, and ten years. The discussion of fund risk is usually based on a subjective assessment of the risk of the assets in the fund.

The conventional approach to presenting statistics on fund performance is inadequate. Funds can be purchased at any time, not just a few arbitrarily selected dates. This post measures the mean and standard deviation of return for two popular funds when there are multiple possible purchase and sale dates for each fund.

Statistical tests are used to evaluate whether the observed difference in return and risk outcomes for two funds are statistically significant.

**Question**: This post considers two of Vanguards most successful funds. VFIAX is a fund that mimics the S&P 500 and VWELX a fund that is around 70% equity and 30% fixed income.

The 48 potential purchase dates for both of the two funds are the first day of each month starting in January 2002 and ending in December of 2005.

The 48 potential sale dates for the two funds are the first day of each month starting in January 2012 and ending in December of 2015.

- Assume that each combination of purchase and sale dates is equally likely.

- What are the expected return and the standard deviation of return for both funds?

- What are the minimum and maximum returns for each fund?

- Can we reject the hypothesis of identical variances for the two funds?

- Can we reject the hypothesis the mean returns are identical?

**Analysis**:

There are 2304 (48 x 48) possible (purchase-sale) outcomes. For each of these outcomes I calculate ln(AP_{2}/AP_{1}) where AP_{2} is the adjusted price in the 2012 to 2015 time period and AP_{1} is adjusted price in the 2002 to 2005 time period.

The mean standard deviation, minimum, and maximum for the two funds are presented below.

Returns from Two Funds |
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Fund Description | Mean | Standard Deviation | Minimum | Maximum | |

VWELX |
Stocks and bonds | 0.763 | 0.177 | 0.341 | 1.14 |

VFIAX |
Stocks | 0.692 | 0.226 | 0.178 | 1.21 |

Sample size 2304 based on 48 possible purchase dates between 2002 and 2005 and 48 possible sale dates 2012 and 2015.

**Observations**:

- The mean return of the bond/stock fund is higher than the mean return of the stock-only fund by around 10 percent.

- The standard deviation of returns for the bond/stock fund is lower than the standard deviation of returns for the stock-only fund by around 21 percent.

- The maximum return is higher for the stock-only fund by around 92 percent.

- The minimum return is lower for the stock-only fund by around 6 percent.

**Comments**:

**Comment One**: The finding that the combined stock/bond fund has a larger mean return compared to the stock-only fund is extremely unusual because over long periods stocks tend to have higher returns than bonds. However, the stock portfolios of the two funds differ. The stock portfolio in VWELX is broadly diversified but does not track a specific index. The stock portfolio in VFIAX tracks the S&P 500. VWELX was able to get higher returns than VFIAX because its stock portfolio outperformed the S&P 500 while the bond portfolio lowered risk. It also did not hurt that interest rates fell and bond prices rose in this time period.

**Comment Two**: The stock-only portfolio was much more risky than the combined bond-stock portfolio. This is evidenced both by the lower standard deviation and the higher minimum return. The minimum return statistic measures the worst-outcome return. The worst-outcome return for the combined stock-bond portfolio is around 92 percent higher than the worst-outcome return for the stock-only portfolio.

**Tests of equal variances for returns:**

A test of the hypothesis that the variances of return for the two portfolios are equal was conducted. The F-statistic comparing the ratio of the two standard deviations was 1.63, which is significantly different from 1.0. The hypothesis that the two variances are identical is rejected.

**Tests of equal mean returns**:

A test of the hypothesis that the mean returns for the two portfolios are equal was conducted. The t-statistic for this hypothesis test was 12.9. The hypothesis of identical means is rejected.

**Technical Note:** I used STATA to make the calculations in this note. Period one and period two data were placed in separate data sets. The N to N merge provides the 2304 outcomes.

**Concluding Thought**: The practice of presenting return numbers on investment funds for a few arbitrarily chosen holding periods is, in my view, not very useful. The holding periods are arbitrary and subject to manipulation. There is no measure of risk.

The technique presented here relies on many possible outcomes defined by different purchase and sale dates. The multiple outcome approach allows for the presentation of risk measures.

The note shows that the performance of the VWELX fund was exceptional in this period.

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