IF such outperformance isn’t enough to overcome the drag of expenses, what would do the trick? Mr. Kritzman calculates that just to break even with the index fund, net of all expenses, the actively managed fund would have to outperform it by an average of 4.3 percentage points a year on a pre-expense basis. For the hedge fund, that margin would have to be 10 points a year.

The chances of finding such funds are next to zero, said Russell Wermers, a finance professor at the University of Maryland. Consider the 452 domestic equity mutual funds in the Morningstar database that existed for the 20 years through January of this year. Morningstar reports that just 13 of those funds beat the Standard & Poor’s 500-stock index by at least four percentage points a year, on average, over that period. That’s less than 3 out of every 100 funds.

But even that sobering statistic paints too rosy a picture, the professor said. That’s because it’s one thing to learn, after the fact, that a fund has done that well, and quite another to identify it in advance. Indeed, he said, he has found from his research that only a minority of funds that beat the market in a given year can outperform it the next year as well.

Professor Wermers said he believed that it was “exceedingly probable that any fund that has beaten the market by an average of more than one percentage point per year over the last decade achieved that return almost entirely due to luck alone.”

“By definition, therefore, such a fund could not have been identified in advance,” he added.

The investment implication is clear, according to Mr. Kritzman. “It is very hard, if not impossible,” he wrote in his study, “to justify active management for most individual, taxable investors, if their goal is to grow wealth.” And he said that those who still insist on an actively managed fund are almost certainly “deluding themselves.”

What if you’re investing in a tax-sheltered account, like a 401(k) or an I.R.A.? In that case, Mr. Kritzman conceded, the odds are relatively more favorable for active management, because, in his simulations, taxes accounted for about two-thirds of the expenses of the actively managed mutual fund and nearly half of the hedge fund’s. But he emphasized the word “relatively.”

“Even in a tax-sheltered account,” he said, “the odds of beating the index fund are still quite poor.”