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The first thing a statistician notices about the world of finance is that common statistics are thrown around in very serious ways without any hint of their statistical significance.
Alpha, beta, or even simple excess benchmark performance feature in official reports but lack supporting t-stats, p-values, significance levels, or confidence intervals. At best, regression R-Squared is present, but even then it often lacks the F-statistic which takes into consideration degrees of freedom. It’s enough to puzzle any statistician, especially because failing to consider these elements can have serious implications.
Case in point the $2.4B AUM American Century Value Fund. As we mentioned in our recent piece, A Closet Index Carol, the fund is the target of a class action lawsuit alleging that they operated as a “closet indexer”. The fund’s underperformance, and in particular its negative alpha against the benchmark, is among the key arguments of the complaint.
Specifically, based on the three-year period October 2018 – September 2021 it is stated that:
- The fund’s 97% R-Squared means that “the strength of the relationship between the Fund and its benchmark, the Russell 1000 Value Index, is very high. Such a strong relationship indicates that the investment advisor’s investment strategy is adding little or no value to the Fund’s performance.”
- The fund’s negative alpha (-1.57% for the period) “further illustrates that the investment advisor’s investment strategy is adding no value to the Fund’s performance.”
These numbers match up with our own analysis from the fund’s Investor shares (TWVLX) return data from Morningstar. But as any statistician can tell you, there are additional data points required to draw conclusions, including the standard ones listed in the table below:
| Alpha, % | -1.46 |
| Alpha t-stat | -0.65 |
| Alpha p-value, % | 52.22 |
| Beta | 1.10 |
| Beta t-stat | 34.06 |
| Beta p-value, % | 0.00 |
| R-Squared, % | 97.15 |
Our alpha1 and R2 are similar to the ones in the complaint; the differences probably come from an alternate share class being used (or some other difference in returns), and possibly annualization type (we used arithmetic).
What is apparent, though, is that both the t-statistic (-0.65) and its p-value (52%) point to a value of alpha that is not statistically significantly different from zero with a 95% confidence interval being (-6.1%,3.1%). Any statistician looking at these numbers would reject this (seemingly significant!) alpha as totally random. In fact, in the peer group of funds with $1B AUM or greater, this fund has some of the most random values for alpha.2
In the scatter chart below, for each fund in the group we show the fund’s annual alpha (Y-axis) vs. its statistical significance or p-value (X-axis). The lower the p-value, the more significantly the fund’s alpha is different from zero. American Century Value fund appears in the half of the group with the least significant values of alpha. It’s worth noting that six funds in the shaded area produced alpha (in the three-year period) that is highly significant at the 0.05 level – positive for three funds and negative for the other three funds.
Low significance could mean that the fund’s return randomly fluctuates around the benchmark, and it could outperform in one period or underperform in another. That’s exactly what we see in the chart below showing trailing 1, 2, and 3-year returns for TWVLX and the peer group above vs. the benchmark – Russell 1000 Value index (as of September 2021). The fund actually handily outperformed its peer group and the benchmark by 6% and 7%, respectively, in the trailing year leading to the lawsuit, albeit slightly underperforming the benchmark over the three-year period.
It’s worth noting though that for the trailing two-year period the fund’s annual performance (14.2%) almost matched the median return of the group (14.3%).
We also looked at American Century Value fund in the context of the two funds from the group of six in the scatter chart above – one having statistically significant negative alpha and the other one statistically significant positive alpha – and that’s how we label them for the chart below. The chart shows rolling 24-month annualized performance for all three vs. the benchmark (in grey).
Note that all three are “hugging” the benchmark in the sense that their twists and turns mimic that of the benchmark. For what’s it worth, the American Century fund is actually deviating more than others in the pattern because its R-squared to the benchmark is not as high as that of the other two funds. Ironic, since high R-squared was another indication of “closet indexing” brought in the complaint.
At the same time, one fund is constantly and steadily creating value above the benchmark while the other is consistently “subtracting” about 4% annually (which is above any imaginable fees). So, they may all be “huggers” in the sense that they do their risk management job well, but clearly one of these funds is more concerning than the others.
Compliance officers, regulators, attorneys – no matter who uses statistics, it is critical to understand the importance of credibility values. It’s why we advocate for full, robust quantitative analysis. After all, statistics get scrutinized in sports and in our daily vaccine/drug efficiency news updates – why wouldn’t people give their investments the same amount of attention?
Edited on 1/16/2022: Added confidence interval and footnotes.
DISCLAIMER:
MPI conducts performance-based analyses and, beyond any public information, does not claim to know or insinuate what the actual strategy, positions or holdings of the funds discussed are, nor are we commenting on the quality or merits of the strategies. This analysis is purely returns-based and does not reflect actual holdings. Deviations between our analysis and the actual holdings and/or management decisions made by funds are expected and inherent in any quantitative analysis. MPI makes no warranties or guarantees as to the accuracy of this statistical analysis, nor does it take any responsibility for investment decisions made by any parties based on this analysis.
Footnotes
- 1Similar to the Complaint, we compute alpha, beta and R2 against the fund’s benchmark – Russell 1000 Value Index.
- 2In the sense of having the widest confidence interval around zero. At the same time, the fund’s beta of 1.1 is highly significant with t-stat of 34 and p-value lower than 1%. This means that the fund has higher risk than the benchmark and is not exactly an “index hugger” – which we discovered through style analysis in our previous blog.