I’ve believed for some time that a broad cap-weighted index fund was virtually guaranteed to beat most investors, as it is guaranteed to beat the average dollar in the market.
That belief is based on arithmetic simply explained in a very short paper, The Arithmetic of Active Management by Nobel Laureate William Sharpe.
“Before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar,” Sharpe wrote. “After costs, the return on the average actively managed dollar will be less than the return on the average passively managed dollar.”
By that logic, an ultra-low-cost total stock index fund like the Vanguard Total Stock Index Fund (VTI) will almost certainly best its peers. Active funds hold more cash, which was among the best-performing assets during the first half of the year, but that wasn’t going to explain a difference of 2 percentage points, with VTI down 21.32% versus the category average down 19.30%.
The explanation had to be one or a combination of three possible explanations: Sharpe’s Arithmetic is wrong, fund managers had a great six months of stock picking, or Morningstar’s benchmark for total stock index funds is imperfect.
Criticism of Sharpe’s Theory
Not everyone has embraced Sharpe’s arithmetic. A 2018 paper in the Financial Analysts Journal pointed out two flaws. First, the stock market is constantly changing, with new issues and stock repurchases, so it’s not a zero-sum game. Secondly, indexes are constantly changing; for example, Tesla’s addition to the S&P 500 in late 2020.
Though these criticisms are valid, are they material?
“The arithmetic must be true in a closed system,” John Rekenthaler, vice president of research at Morningstar, who helped develop the famous Morningstar style box, told me. “The impact of new issues and stock buybacks strikes me as an immaterial wrinkle.”
When I posed the same question to William Sharpe himself recently, his response was that these were “very minor imperfections around the edge.”
I agree with Sharpe and Rekenthaler. Rekenthaler also confirmed there is very little impact from new constituents in a total stock index fund. Overall, Sharpe’s arithmetic holds in every material way.
The Prowess of Active Managers
So, did active fund managers have a great first half of the year? Sharpe acknowledged to me that active fund managers don’t include nonfund investors, and it’s certainly possible that they beat the market.
But there are thousands of fund managers who comprise a significantly large part of the total stock market. Other parts of the stock market have a high concentration of professional managers as well. Statistically, I think there is a very low probability that thousands of fund managers suddenly performed better.
When it comes to benchmarking, as of Aug. 19, 2022, VTI was at the bottom 31% of funds, though the shortfall vs. peers were reduced to a still significant 109 bps.
A total stock index fund can be viewed as generally comprising the holdings of a fund such as the Vanguard 500 ETF (VOO), and a completion fund tracking an extended market index such as the Vanguard Extended Market ETF (VXF), which owns all of the stocks VTI owns except those in the S&P 500.
A ratio of roughly 80% of S&P 500 and 20% of extended market is comparable to the total market. But, for the first half of the year, Morningstar ranked VOO at the 42nd percentile and VXF at the 31st percentile. Morningstar used a large cap blend to benchmark the 500, midcap growth for the extended market, and large cap and midcap for VTI. That’s not consistent.
Morningstar’s methodology for looking at the style of a total stock index fund has it tilted toward growth, and growth badly lagged large and midcap blend through June. Although they had a significant snap-back so far in this quarter, hence the 200 bp shortfall through June that has been cut by more than half since.
“Properly measured, the average actively managed dollar must underperform the average passively managed dollar, net of costs,” Sharpe’s paper noted. “Empirical analyses that appear to refute this principle are guilty of improper measurement.”
Homing in on the Problem
When I ran my explanation of this tilt bias in benchmarking by Rekenthaler, he agreed.
“The style boxes are a simple descriptive way of viewing and benchmarking performance,” he said. “This example points out a drawback on the simplicity.”
I think no one does a better job of benchmarking than Morningstar, and the dilemma is to either create a simple and understandable benchmarking system or something so complex it becomes precisely useless.
“Owning a low-cost market cap-weighted total index fund is both psychologically and mathematically superior,” Rekenthaler concluded. Over 15 years, VTI's performance ranked in the 19th percentile and would have done even better if there were an adjustment for survivorship bias, taking into account the funds that became extinct due to poor performance.
Indeed, when it comes to investing, there aren’t many things we can count on. It’s nice to know arithmetic is still one.
Allan Roth is the founder of Wealth Logic LLC, an hourly based financial planning firm. He is required by law to note that his columns are not meant as specific investment advice. Roth also writes for Barrons, AARP, Advisor Perspectives and Financial Planning magazine. You can reach him at [email protected], or follow him on Twitter at Allan Roth (@Dull_Investing) · Twitter.