William Sharpe

Another Arithmetic Lesson from William Sharpe

William Sharpe

Nobel Laureate William Sharpe is one of the academic linchpins of IFA’s investment philosophy. A year after winning the Nobel Prize in 1990, he penned a brief and elegant article1 showing that as a whole, active investors must do worse than passive investors simply because active investors incur higher costs and the returns before costs are essentially a zero-sum game. In a newly published article2 in the same journal, Sharpe demonstrates just how corrosive these higher costs are on investor wealth. Once again, it is just a question of arithmetic, which is beyond dispute.

Sharpe performs calculations and runs simulations with two hypothetical funds—a high-cost fund with an expense ratio of 1.12% and a low-cost fund with an expense ratio of 0.06%. These numbers were derived from Vanguard’s data on mutual funds. The key point is that both funds have the same expected return before costs. Although the 1.06% difference may not seem like a lot, the cumulative impact over 20 to 30 years is staggering. In the author’s words, “Under plausible conditions, a person saving for retirement who chooses low-cost investments could have a standard of living throughout retirement more than 20% higher than that of a comparable investor in high-cost investments.”

Sharpe asks why, in the face of these dramatic results, so many investors still choose high-cost active funds? Aside from the massive amount of misinformation that pervades the financial media, Sharpe notes that the high-cost active fund still has a small chance of beating the low-cost index fund due to randomness (or luck), so “hope may spring eternal.” However, as we have said so many times at Index Funds Advisors, hope is not a viable investment strategy.

One very important factor not included in Sharpe’s analysis is the destructive impact of investor behavior. Investors who utilize active managers are more likely to engage in returns-chasing and market-timing, and thus may have a more difficult time capturing the time-weighted returns of the funds in which they invest, as shown in the chart below. And Do-it-Yourself Indexers, as we interpret the studies of John Bogle and Morningstar to based on, (click on the source links in chart) have done better than investors in active funds, but still miss out on a substantial share of index fund time weighted returns due to behavioral finance errors. Lastly, according to Morningstar, investors in Dimensional funds who are guided by advisors were the best behaved of the groups. This is the only study that we could find that was published by an independent source on this topic. Please click this link to read the Morningstar study. IFA is in the process of doing an internal study on it's own client's behavior.

For more information on the sources of higher investment costs, please refer to IFA’s Step 7: Silent Partners.


1Sharpe, William F. 1991. “The Arithmetic of Active Management.” Financial Analysts Journal, vol. 47, no. 1 (January/February):7-9.

2Sharpe, William F. 2013. “The Arithmetic of Investment Expenses.” Financial Analysts Journal, vol. 69, no. 2 (March/April):34-41.