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How can one relatively easily show that low expense ratio funds outperform high expense ratio funds?
I hope a wall of text with citations qualifies as "relatively easy." Many of these studies are worth quoting at length. Long story short, a great deal of research has found that actively-managed funds underperform market indexes and passively-managed funds because of their high turnover and higher fees, among other factors.  Longer answer: Chris is right in stating that survivorship bias presents a problem for such research; however, there are several academic papers that address the survivorship problem, as well as the wider subject of active vs. passive performance. I'll try to provide a brief summary of some of the relevant literature.  The seminal paper that started the debate is Michael Jensen's 1968 paper titled "The Performance of Mutual Funds in the Period 1945-1964". This is the paper where Jensen's alpha, the ubiquitous measure of the performance of mutual fund managers, was first defined. Using a dataset of 115 mutual fund managers, Jensen finds that The evidence on mutual fund performance indicates not only that these 115 mutual funds were on average not able to predict security prices well enough to outperform a buy-the-market-and-hold policy, but also that there is very little evidence that any individual fund was able to do significantly better than that which we expected from mere random chance. Although this paper doesn't address problems of survivorship, it's notable because, among other points, it found that managers who actively picked stocks performed worse even when fund expenses were ignored. Since actively-managed funds tend to have higher expenses than passive funds, the actual picture looks even worse for actively managed funds. A more recent paper on the subject, which draws similar conclusions, is Martin Gruber's 1996 paper "Another puzzle: The growth in actively managed mutual funds". Gruber calls it "a puzzle" that investors still invest in actively-managed funds, given that their performance on average has been inferior to that of index funds. He addresses survivorship bias by tracking funds across the entire sample, including through mergers. Since most mutual funds that disappear are merged into existing funds, he assumes that investors in a fund that disappear choose to continue investing their money in the fund that resulted from the merger. Using this assumption and standard measures of mutual fund performance, Gruber finds that mutual funds underperform an appropriately weighted average of the indices by about 65 basis points per year. Expense ratios for my sample averaged 113 basis points a year. These numbers suggest that active management adds value, but that mutual funds charge the investor more than the value added. Another nice paper is Mark Carhart's 1997 paper "On persistence in mutual fund performance" uses a sample free of survivorship bias because it includes "all known equity funds over this period." It's worth quoting parts of this paper in full: I demonstrate that expenses have at least a one-for-one negative impact on fund performance, and that turnover also negatively impacts performance. ... Trading reduces performance by approximately 0.95% of the trade's market value.  In reference to expense ratios and other fees, Carhart finds that The investment costs of expense ratios, transaction costs, and load fees all have a direct, negative impact on performance. The study also finds that funds with abnormally high returns last year usually have higher-than-expected returns next year, but not in the following years, because of momentum effects. Lest you think the news is all bad, Russ Wermer's 2000 study "Mutual fund performance: An empirical decomposition into stock‐picking talent, style, transactions costs, and expenses" provides an interesting result. He finds that many actively-managed mutual funds hold stocks that outperform the market, even though the net return of the funds themselves underperforms passive funds and the market itself.  On a net-return level, the funds underperform broad market indexes by one percent a year. Of the 2.3% difference between the returns on stock holdings and the net returns of the funds, 0.7% per year is due to the lower average returns of the nonstock holdings of the funds during the period (relative to stocks). The remaining 1.6% per year is split almost evenly between the expense ratios and the transaction costs of the funds. The final paper I'll cite is a 2008 paper by Fama and French (of the Fama-French model covered in business schools) titled, appropriately, "Mutual Fund Performance". The paper is pretty technical, and somewhat above my level at this time of night, but the authors state one of their conclusions bluntly quite early on: After costs (that is, in terms of net returns to investors) active investment is a negative sum game.  Emphasis mine. In short, expense ratios, transaction costs, and other fees quickly diminish the returns to active investment. They find that The [value-weight] portfolio of mutual funds that invest primarily in U.S. equities is close to the market portfolio, and estimated before fees and expenses, its alpha is close to zero. Since the [value-weight] portfolio of funds produces an α close to zero in gross returns, the alpha estimated on the net returns to investors is negative by about the amount of fees and expenses. This implies that the higher the fees, the farther alpha decreases below zero. Since actively-managed mutual funds tend to have higher expense ratios than passively-managed index funds, it's safe to say that their net return to the investor is worse than a market index itself. I don't know of any free datasets that would allow you to research this, but one highly-regarded commercial dataset is the CRSP Survivor-Bias-Free US Mutual Fund Database from the Center for Research in Security Prices at the University of Chicago. In financial research, CRSP is one of the "gold standards" for historical market data, so if you can access that data (perhaps for a firm or academic institution, if you're affiliated with one that has access), it's one way you could run some numbers yourself.